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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 $f : {} _ { R } A \rightarrow {} _ { R } R$ 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 [[#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:

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 $f : {} _ { R } A \rightarrow {} _ { R } R$, 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 [[#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 $C$.

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:

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 $0 \neq A , B \lhd  R$, $f : {} _ { R } A \rightarrow {} _ { R } R$, $g : B _ { R } \rightarrow R _ { R }$ 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 [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a8]]], [[#References|[a13]]].

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]]].

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:

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 $f : {} _ { R } A \rightarrow {} _ { R } R$, 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====
<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>