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This ring of quotients was introduced in [[#References|[a6]]] as a tool to study prime rings satisfying a generalized polynomial identity. Specifically, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201201.png" /> be a [[Prime ring|prime ring]] (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201202.png" />) and consider all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201204.png" /> is a non-zero ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201205.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201206.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201207.png" />-module mapping. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201209.png" /> are equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012012.png" /> agree on their common domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012013.png" />. This is easily seen to yield an equivalence relation, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012014.png" /> of all equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012015.png" /> is a ring extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012016.png" /> with arithmetic defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012017.png" /></td> </tr></table>
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This ring of quotients was introduced in [[#References|[a6]]] as a tool to study prime rings satisfying a generalized polynomial identity. Specifically, let $R$ be a [[Prime ring|prime ring]] (with $1$) and consider all pairs $( A , f )$, where $A$ is a non-zero ideal of $R$ and where $f : \square _ { R } A \rightarrow \square _ { R } R$ is a left $R$-module mapping. One says that $( A , f )$ and $( A ^ { \prime } , f ^ { \prime } )$ are equivalent if $f$ and $f ^ { \prime }$ agree on their common domain $A \cap A ^ { \prime }$. This is easily seen to yield an equivalence relation, and the set $Q_{\text{l}} ( R )$ of all equivalence classes $[ A , f ]$ is a ring extension of $R$ with arithmetic defined by
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012019.png" /> indicates the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012020.png" /> followed by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012021.png" />.
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\begin{equation*} [ A , f ] + [ B , g ] = [ A \bigcap B , f + g ], \end{equation*}
  
One can show (see [[#References|[a12]]]) that the left Martindale ring of quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012022.png" /> is characterized as the unique (up to isomorphism) ring extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012024.png" /> satisfying:
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\begin{equation*} [A,f ] [ B , g ] = [ B A , f g ]. \end{equation*}
  
1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012025.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012027.png" />;
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Here, $f g$ indicates the mapping $f$ followed by the mapping $g$.
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012030.png" />; and
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One can show (see [[#References|[a12]]]) that the left Martindale ring of quotients $Q_{\text{l}} ( R )$ is characterized as the unique (up to isomorphism) ring extension $Q$ of $R$ satisfying:
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012032.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012033.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012035.png" />. As a consequence, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012036.png" /> is simple, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012037.png" />. In any case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012038.png" /> is certainly a prime ring. The right Martindale ring of quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012039.png" /> is defined in an analogous manner and enjoys similar properties.
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1) if $q \in Q$, then there exists a $0 \neq A \lhd  R$ with $A q \subseteq R$;
  
Again, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012040.png" /> be a prime ring and write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012042.png" /> is a field known as the extended centroid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012043.png" />, and the subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012045.png" /> is called the central closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012046.png" />. One can show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012047.png" /> is a prime ring which is centrally closed, namely it contains its extended centroid. This central closure controls the linear identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012048.png" /> in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012049.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012050.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012051.png" />, then there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012052.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012054.png" />. Martindale's theorem [[#References|[a6]]] asserts that a prime ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012055.png" /> satisfies a non-trivial generalized polynomial identity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012056.png" /> has an idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012058.png" /> is a minimal right [[Ideal|ideal]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012059.png" /> is a [[Division algebra|division algebra]] that is finite dimensional over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012060.png" />.
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2) if $0 \neq q \in Q$ and $0 \neq I \triangleleft R$, then $Iq  \neq 0$; and
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012061.png" /> is a non-commutative [[Free algebra|free algebra]] in two variables, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012062.png" /> is a domain but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012063.png" /> is not. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012064.png" /> is in some sense too large an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012065.png" />. In [[#References|[a3]]], it was suggested that for any prime ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012066.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012067.png" /> would define a symmetric version of the Martindale ring of quotients. This was shown to be the case in [[#References|[a12]]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012068.png" /> was characterized as the unique (up to isomorphism) ring extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012070.png" /> satisfying:
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3) if $0 \neq A \lhd  R$ and $f : \square _ { R } A \rightarrow \square _ { R } R$, then there exists a $q \in Q$ with $a f = a q$ for all $a \in A$. As a consequence, if $R$ is simple, then $Q _ { \text{l} } ( R ) = R$. In any case, $Q_{\text{l}} ( R )$ is certainly a prime ring. The right Martindale ring of quotients $Q _ { r } ( R )$ is defined in an analogous manner and enjoys similar properties.
  
a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012071.png" />, then there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012072.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012073.png" />;
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Again, let $R$ be a prime ring and write $Q = Q _ { \text{l} } ( R )$. Then $C = \mathbf{Z} ( R ) = \mathbf{C} _ { Q } ( R )$ is a field known as the extended centroid of $R$, and the subring $R C$ of $Q$ is called the central closure of $R$. One can show that $R C$ is a prime ring which is centrally closed, namely it contains its extended centroid. This central closure controls the linear identities of $R$ in the sense that if $0 \neq a , b , c , d \in R$ with $axb=cxd$ for all $x \in R$, then there exists an element $0 \neq q \in C$ with $c = a q$ and $d = q ^ { - 1 } b$. Martindale's theorem [[#References|[a6]]] asserts that a prime ring $R$ satisfies a non-trivial generalized polynomial identity if and only if $R C$ has an idempotent $e$ such that $e R C$ is a minimal right [[Ideal|ideal]] and $eR Ce$ is a [[Division algebra|division algebra]] that is finite dimensional over $C$.
  
b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012076.png" />; and
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If $R = F \langle x , y \rangle$ is a non-commutative [[Free algebra|free algebra]] in two variables, then $R$ is a domain but $Q_{\text{l}} ( R )$ is not. Thus $Q_{\text{l}} ( R )$ is in some sense too large an extension of $R$. In [[#References|[a3]]], it was suggested that for any prime ring $R$, the set $Q _ { s } ( R ) = \{ q \in Q_{\text{l} } ( R ) : q B \subseteq R \ \text { for some } \ 0 \neq B \lhd R \}$ would define a symmetric version of the Martindale ring of quotients. This was shown to be the case in [[#References|[a12]]], where $Q _ { s } ( R )$ was characterized as the unique (up to isomorphism) ring extension $Q$ of $R$ satisfying:
  
c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012080.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012082.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012083.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012085.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012087.png" />.
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a) if $q \in Q$, then there exist $0 \neq A , B \lhd  R$ with $A q , q B \subseteq R$;
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012088.png" /> is a domain, then so is its symmetric Martindale ring of quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012089.png" />. Furthermore, any non-commutative free algebra is symmetrically closed.
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b) if $0 \neq q \in Q$ and $0 \neq I \triangleleft R$, then $I q , q I \neq 0$; and
  
An interesting example here is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012090.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012091.png" />-vector space of all square matrices of some infinite size, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012092.png" /> be the subspace which is the direct sum of the scalar matrices and the matrices with only finitely many non-zero entries. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012093.png" /> is a prime ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012094.png" /> is the ring of row-finite matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012096.png" /> is the ring of column-finite matrices, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012097.png" /> is the ring consisting of matrices which are both row and column finite. Thus, in some rough sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012098.png" /> is the intersection of the left and right Martindale rings of quotients. Other examples of interest can be found in [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a8]]], [[#References|[a13]]].
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c) if $0 \neq A , B \lhd  R$, $f : \square _ { R } A \rightarrow \square _ { R } R$, $g : B _ { R } \rightarrow R _ { R }$ and $( a f ) b = a ( g b )$ for all $a \in A$, $b \in B$, then there exists a $q \in Q$ with $a f = a q$ and $g b = q  b $ for all $a \in A$, $b \in B$.
  
Another important intermediate ring is the normal closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012099.png" />, defined in [[#References|[a10]]] as the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120100.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120101.png" /> is the multiplicatively closed set of all units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120103.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120105.png" /> is the smallest ring extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120106.png" /> needed to study all group actions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120107.png" />. Despite its name, the normal closure is not necessarily normally closed. Again, numerous examples of these normal closures have been computed. See, for example, [[#References|[a7]]], [[#References|[a9]]], [[#References|[a11]]].
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When $R$ is a domain, then so is its symmetric Martindale ring of quotients $Q _ { s } ( R )$. Furthermore, any non-commutative free algebra is symmetrically closed.
  
Finally, as was pointed out in [[#References|[a1]]], there is a more general construction which yields analogues of the Martindale ring of quotients for rings which are not necessarily prime. To this end, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120108.png" /> be an arbitrary ring (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120109.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120110.png" /> be a non-empty [[Filter|filter]] of ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120111.png" />. Specifically, it is assumed that:
+
An interesting example here is as follows. Let $\mathcal{M} _ { \infty } ( F )$ denote the $F$-vector space of all square matrices of some infinite size, and let $R$ be the subspace which is the direct sum of the scalar matrices and the matrices with only finitely many non-zero entries. Then $R$ is a prime ring, $Q_{\text{l}} ( R )$ is the ring of row-finite matrices in $\mathcal{M} _ { \infty } ( F )$, $Q _ { r } ( R )$ is the ring of column-finite matrices, and $Q _ { s } ( R )$ is the ring consisting of matrices which are both row and column finite. Thus, in some rough sense, $Q _ { s } ( R )$ is the intersection of the left and right Martindale rings of quotients. Other examples of interest can be found in [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a8]]], [[#References|[a13]]].
  
every ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120112.png" /> is regular, that is, has trivial right and left annihilator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120113.png" />;
+
Another important intermediate ring is the normal closure of $R$, defined in [[#References|[a10]]] as the product $R N$, where $N$ is the multiplicatively closed set of all units $u \in Q _ { \text{l} } ( R )$ with $u ^ { - 1 } R u = R$. Then $R C \subseteq R N \subseteq Q _ { s } ( R )$, and $R N$ is the smallest ring extension of $R$ needed to study all group actions on $R$. Despite its name, the normal closure is not necessarily normally closed. Again, numerous examples of these normal closures have been computed. See, for example, [[#References|[a7]]], [[#References|[a9]]], [[#References|[a11]]].
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120114.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120115.png" />; and
+
Finally, as was pointed out in [[#References|[a1]]], there is a more general construction which yields analogues of the Martindale ring of quotients for rings which are not necessarily prime. To this end, let $R$ be an arbitrary ring (with $1$) and let $\mathcal{F}$ be a non-empty [[Filter|filter]] of ideals of $R$. Specifically, it is assumed that:
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120116.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120117.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120118.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120119.png" />. Given such a filter, one can again consider all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120120.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120121.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120122.png" />, and use these to construct a ring extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120123.png" /> which might be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120124.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120125.png" /> is a [[semi-prime ring]], then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120126.png" /> of all regular ideals is such a filter. Here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120127.png" />, then the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120128.png" /> is no longer a [[Field|field]], in general, but it is at least a commutative [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]]. Another example of interest occurs when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120129.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120131.png" />-prime ring, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120132.png" /> is a fixed group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120133.png" />. In this case, one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120134.png" /> to be the set of non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120135.png" />-stable ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120136.png" />, and then the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120137.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120138.png" /> extends to an action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120139.png" />.
+
every ideal $A \in \mathcal{F}$ is regular, that is, has trivial right and left annihilator in $R$;
 +
 
 +
if $A , B \in \cal{F}$, then $A B \in \mathcal{F}$; and
 +
 
 +
if $A \in \mathcal{F}$ and if $B \triangleleft R$ with $A \subseteq B$, then $B \in \mathcal{F}$. Given such a filter, one can again consider all pairs $( A , f )$ with $A \in \mathcal{F}$ and with $f : \square _ { R } A \rightarrow \square _ { R } R$, and use these to construct a ring extension of $R$ which might be denoted by $Q _ { \mathcal{F} } ( R )$. For example, if $R$ is a [[semi-prime ring]], then the set $\mathcal{F}$ of all regular ideals is such a filter. Here, if $Q = Q _ { \mathcal{F} } ( R )$, then the centre $C = \mathbf{Z} ( Q )$ is no longer a [[Field|field]], in general, but it is at least a commutative [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]]. Another example of interest occurs when $R$ is a $G$-prime ring, where $G$ is a fixed group of automorphisms of $R$. In this case, one can take $\mathcal{F}$ to be the set of non-zero $G$-stable ideals of $R$, and then the action of $G$ on $R$ extends to an action on $Q _ { \mathcal{F} } ( R )$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.A. Amitsur,  "On rings of quotients" , ''Symposia Math.'' , '''VIII''' , Acad. Press  (1972)  pp. 149–164</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Ara,  A. del Rio,  "A question of Passman on the symmetric ring of quotients"  ''Israel J. Math.'' , '''68'''  (1989)  pp. 348–352</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.K. Kharchenko,  "Generalized identities with automorphisms"  ''Algebra and Logic'' , '''14'''  (1976)  pp. 132–148  ''Algebra i Logika'' , '''14'''  (1975)  pp. 215–237</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.K. Kharchenko,  "Algebras of invariants of free algebras"  ''Algebra and Logic'' , '''17'''  (1979)  pp. 316–321  ''Algebra i Logika'' , '''17'''  (1978)  pp. 478–487</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Lewin,  "The symmetric ring of quotients of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120140.png" />-fir"  ''Commun. Algebra'' , '''16'''  (1988)  pp. 1727–1732</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W.S. Martindale III,  "Prime rings satisfying a generalized polynomial identity"  ''J. Algebra'' , '''12'''  (1969)  pp. 576–584</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W.S. Martindale III,  "The normal closure of the coproduct of rings over a division ring"  ''Trans. Amer. Math. Soc.'' , '''293'''  (1986)  pp. 303–317</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W.S. Martindale III,  "The symmetric ring of quotients of the coproduct of rings"  ''J. Algebra'' , '''143'''  (1991)  pp. 295–306</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W.S. Martindale III,  S. Montgomery,  "The normal closure of coproducts of domains"  ''J. Algebra'' , '''82'''  (1983)  pp. 1–17</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S. Montgomery,  "Automorphism groups of rings with no nilpotent elements"  ''J. Algebra'' , '''60'''  (1979)  pp. 238–248</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  S. Montgomery,  "X-inner automorphisms of filtered algebras"  ''Proc. Amer. Math. Soc.'' , '''83'''  (1981)  pp. 263–268</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D.S. Passman,  "Computing the symmetric ring of quotients"  ''J. Algebra'' , '''105'''  (1987)  pp. 207–235</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  J.D. Rosen,  M.P. Rosen,  "The Martindale ring of quotients of a skew polynomial ring of automorphism type"  ''Commun. Algebra'' , '''21'''  (1993)  pp. 4051–4063</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  S.A. Amitsur,  "On rings of quotients" , ''Symposia Math.'' , '''VIII''' , Acad. Press  (1972)  pp. 149–164</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P. Ara,  A. del Rio,  "A question of Passman on the symmetric ring of quotients"  ''Israel J. Math.'' , '''68'''  (1989)  pp. 348–352</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  V.K. Kharchenko,  "Generalized identities with automorphisms"  ''Algebra and Logic'' , '''14'''  (1976)  pp. 132–148  ''Algebra i Logika'' , '''14'''  (1975)  pp. 215–237</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V.K. Kharchenko,  "Algebras of invariants of free algebras"  ''Algebra and Logic'' , '''17'''  (1979)  pp. 316–321  ''Algebra i Logika'' , '''17'''  (1978)  pp. 478–487</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J. Lewin,  "The symmetric ring of quotients of a $2$-fir"  ''Commun. Algebra'' , '''16'''  (1988)  pp. 1727–1732</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  W.S. Martindale III,  "Prime rings satisfying a generalized polynomial identity"  ''J. Algebra'' , '''12'''  (1969)  pp. 576–584</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  W.S. Martindale III,  "The normal closure of the coproduct of rings over a division ring"  ''Trans. Amer. Math. Soc.'' , '''293'''  (1986)  pp. 303–317</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  W.S. Martindale III,  "The symmetric ring of quotients of the coproduct of rings"  ''J. Algebra'' , '''143'''  (1991)  pp. 295–306</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  W.S. Martindale III,  S. Montgomery,  "The normal closure of coproducts of domains"  ''J. Algebra'' , '''82'''  (1983)  pp. 1–17</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  S. Montgomery,  "Automorphism groups of rings with no nilpotent elements"  ''J. Algebra'' , '''60'''  (1979)  pp. 238–248</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S. Montgomery,  "X-inner automorphisms of filtered algebras"  ''Proc. Amer. Math. Soc.'' , '''83'''  (1981)  pp. 263–268</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  D.S. Passman,  "Computing the symmetric ring of quotients"  ''J. Algebra'' , '''105'''  (1987)  pp. 207–235</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  J.D. Rosen,  M.P. Rosen,  "The Martindale ring of quotients of a skew polynomial ring of automorphism type"  ''Commun. Algebra'' , '''21'''  (1993)  pp. 4051–4063</td></tr></table>

Revision as of 17:01, 1 July 2020

This ring of quotients was introduced in [a6] as a tool to study prime rings satisfying a generalized polynomial identity. Specifically, let $R$ be a prime ring (with $1$) and consider all pairs $( A , f )$, where $A$ is a non-zero ideal of $R$ and where $f : \square _ { R } A \rightarrow \square _ { R } R$ is a left $R$-module mapping. One says that $( A , f )$ and $( A ^ { \prime } , f ^ { \prime } )$ are equivalent if $f$ and $f ^ { \prime }$ agree on their common domain $A \cap A ^ { \prime }$. This is easily seen to yield an equivalence relation, and the set $Q_{\text{l}} ( R )$ of all equivalence classes $[ A , f ]$ is a ring extension of $R$ with arithmetic defined by

\begin{equation*} [ A , f ] + [ B , g ] = [ A \bigcap B , f + g ], \end{equation*}

\begin{equation*} [A,f ] [ B , g ] = [ B A , f g ]. \end{equation*}

Here, $f g$ indicates the mapping $f$ followed by the mapping $g$.

One can show (see [a12]) that the left Martindale ring of quotients $Q_{\text{l}} ( R )$ is characterized as the unique (up to isomorphism) ring extension $Q$ of $R$ satisfying:

1) if $q \in Q$, then there exists a $0 \neq A \lhd R$ with $A q \subseteq R$;

2) if $0 \neq q \in Q$ and $0 \neq I \triangleleft R$, then $Iq \neq 0$; and

3) if $0 \neq A \lhd R$ and $f : \square _ { R } A \rightarrow \square _ { R } R$, then there exists a $q \in Q$ with $a f = a q$ for all $a \in A$. As a consequence, if $R$ is simple, then $Q _ { \text{l} } ( R ) = R$. In any case, $Q_{\text{l}} ( R )$ is certainly a prime ring. The right Martindale ring of quotients $Q _ { r } ( R )$ is defined in an analogous manner and enjoys similar properties.

Again, let $R$ be a prime ring and write $Q = Q _ { \text{l} } ( R )$. Then $C = \mathbf{Z} ( R ) = \mathbf{C} _ { Q } ( R )$ is a field known as the extended centroid of $R$, and the subring $R C$ of $Q$ is called the central closure of $R$. One can show that $R C$ is a prime ring which is centrally closed, namely it contains its extended centroid. This central closure controls the linear identities of $R$ in the sense that if $0 \neq a , b , c , d \in R$ with $axb=cxd$ for all $x \in R$, then there exists an element $0 \neq q \in C$ with $c = a q$ and $d = q ^ { - 1 } b$. Martindale's theorem [a6] asserts that a prime ring $R$ satisfies a non-trivial generalized polynomial identity if and only if $R C$ has an idempotent $e$ such that $e R C$ is a minimal right ideal and $eR Ce$ is a division algebra that is finite dimensional over $C$.

If $R = F \langle x , y \rangle$ is a non-commutative free algebra in two variables, then $R$ is a domain but $Q_{\text{l}} ( R )$ is not. Thus $Q_{\text{l}} ( R )$ is in some sense too large an extension of $R$. In [a3], it was suggested that for any prime ring $R$, the set $Q _ { s } ( R ) = \{ q \in Q_{\text{l} } ( R ) : q B \subseteq R \ \text { for some } \ 0 \neq B \lhd R \}$ would define a symmetric version of the Martindale ring of quotients. This was shown to be the case in [a12], where $Q _ { s } ( R )$ was characterized as the unique (up to isomorphism) ring extension $Q$ of $R$ satisfying:

a) if $q \in Q$, then there exist $0 \neq A , B \lhd R$ with $A q , q B \subseteq R$;

b) if $0 \neq q \in Q$ and $0 \neq I \triangleleft R$, then $I q , q I \neq 0$; and

c) if $0 \neq A , B \lhd R$, $f : \square _ { R } A \rightarrow \square _ { R } R$, $g : B _ { R } \rightarrow R _ { R }$ and $( a f ) b = a ( g b )$ for all $a \in A$, $b \in B$, then there exists a $q \in Q$ with $a f = a q$ and $g b = q b $ for all $a \in A$, $b \in B$.

When $R$ is a domain, then so is its symmetric Martindale ring of quotients $Q _ { s } ( R )$. Furthermore, any non-commutative free algebra is symmetrically closed.

An interesting example here is as follows. Let $\mathcal{M} _ { \infty } ( F )$ denote the $F$-vector space of all square matrices of some infinite size, and let $R$ be the subspace which is the direct sum of the scalar matrices and the matrices with only finitely many non-zero entries. Then $R$ is a prime ring, $Q_{\text{l}} ( R )$ is the ring of row-finite matrices in $\mathcal{M} _ { \infty } ( F )$, $Q _ { r } ( R )$ is the ring of column-finite matrices, and $Q _ { s } ( R )$ is the ring consisting of matrices which are both row and column finite. Thus, in some rough sense, $Q _ { s } ( R )$ is the intersection of the left and right Martindale rings of quotients. Other examples of interest can be found in [a2], [a4], [a5], [a8], [a13].

Another important intermediate ring is the normal closure of $R$, defined in [a10] as the product $R N$, where $N$ is the multiplicatively closed set of all units $u \in Q _ { \text{l} } ( R )$ with $u ^ { - 1 } R u = R$. Then $R C \subseteq R N \subseteq Q _ { s } ( R )$, and $R N$ is the smallest ring extension of $R$ needed to study all group actions on $R$. Despite its name, the normal closure is not necessarily normally closed. Again, numerous examples of these normal closures have been computed. See, for example, [a7], [a9], [a11].

Finally, as was pointed out in [a1], there is a more general construction which yields analogues of the Martindale ring of quotients for rings which are not necessarily prime. To this end, let $R$ be an arbitrary ring (with $1$) and let $\mathcal{F}$ be a non-empty filter of ideals of $R$. Specifically, it is assumed that:

every ideal $A \in \mathcal{F}$ is regular, that is, has trivial right and left annihilator in $R$;

if $A , B \in \cal{F}$, then $A B \in \mathcal{F}$; and

if $A \in \mathcal{F}$ and if $B \triangleleft R$ with $A \subseteq B$, then $B \in \mathcal{F}$. Given such a filter, one can again consider all pairs $( A , f )$ with $A \in \mathcal{F}$ and with $f : \square _ { R } A \rightarrow \square _ { R } R$, and use these to construct a ring extension of $R$ which might be denoted by $Q _ { \mathcal{F} } ( R )$. For example, if $R$ is a semi-prime ring, then the set $\mathcal{F}$ of all regular ideals is such a filter. Here, if $Q = Q _ { \mathcal{F} } ( R )$, then the centre $C = \mathbf{Z} ( Q )$ is no longer a field, in general, but it is at least a commutative regular ring (in the sense of von Neumann). Another example of interest occurs when $R$ is a $G$-prime ring, where $G$ is a fixed group of automorphisms of $R$. In this case, one can take $\mathcal{F}$ to be the set of non-zero $G$-stable ideals of $R$, and then the action of $G$ on $R$ extends to an action on $Q _ { \mathcal{F} } ( R )$.

References

[a1] S.A. Amitsur, "On rings of quotients" , Symposia Math. , VIII , Acad. Press (1972) pp. 149–164
[a2] P. Ara, A. del Rio, "A question of Passman on the symmetric ring of quotients" Israel J. Math. , 68 (1989) pp. 348–352
[a3] V.K. Kharchenko, "Generalized identities with automorphisms" Algebra and Logic , 14 (1976) pp. 132–148 Algebra i Logika , 14 (1975) pp. 215–237
[a4] V.K. Kharchenko, "Algebras of invariants of free algebras" Algebra and Logic , 17 (1979) pp. 316–321 Algebra i Logika , 17 (1978) pp. 478–487
[a5] J. Lewin, "The symmetric ring of quotients of a $2$-fir" Commun. Algebra , 16 (1988) pp. 1727–1732
[a6] W.S. Martindale III, "Prime rings satisfying a generalized polynomial identity" J. Algebra , 12 (1969) pp. 576–584
[a7] W.S. Martindale III, "The normal closure of the coproduct of rings over a division ring" Trans. Amer. Math. Soc. , 293 (1986) pp. 303–317
[a8] W.S. Martindale III, "The symmetric ring of quotients of the coproduct of rings" J. Algebra , 143 (1991) pp. 295–306
[a9] W.S. Martindale III, S. Montgomery, "The normal closure of coproducts of domains" J. Algebra , 82 (1983) pp. 1–17
[a10] S. Montgomery, "Automorphism groups of rings with no nilpotent elements" J. Algebra , 60 (1979) pp. 238–248
[a11] S. Montgomery, "X-inner automorphisms of filtered algebras" Proc. Amer. Math. Soc. , 83 (1981) pp. 263–268
[a12] D.S. Passman, "Computing the symmetric ring of quotients" J. Algebra , 105 (1987) pp. 207–235
[a13] J.D. Rosen, M.P. Rosen, "The Martindale ring of quotients of a skew polynomial ring of automorphism type" Commun. Algebra , 21 (1993) pp. 4051–4063
How to Cite This Entry:
Martindale ring of quotients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Martindale_ring_of_quotients&oldid=50388
This article was adapted from an original article by D.S. Passman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article