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''fir.''
 
''fir.''
  
A (non-commutative) ring (with unit element) in which all (one-sided) ideals are free. More precisely, a right fir is a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415801.png" /> in which all right ideals are free of unique rank, as right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415802.png" />-modules. A left fir is defined correspondingly. Firs may be regarded as generalizing the notion of a principal ideal domain.
+
A (non-commutative) ring (with unit element) in which all (one-sided) ideals are free. More precisely, a right fir is a ring $  R $
 +
in which all right ideals are free of unique rank, as right $  R $-
 +
modules. A left fir is defined correspondingly. Firs may be regarded as generalizing the notion of a principal ideal domain.
  
Consider dependence relations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415804.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415805.png" /> a row vector, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415806.png" /> a column vector). Such a relation is called trivial if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415807.png" /> either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415808.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f0415809.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158010.png" />-term relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158011.png" /> is trivialized by an invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158012.png" /> matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158013.png" /> if the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158014.png" /> is trivial. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158015.png" /> be a non-zero ring with unit element, then the following properties are all equivalent: i) every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158016.png" />-term relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158018.png" />, can be trivialized by an invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158019.png" /> matrix; ii) given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158021.png" />, which are right linearly dependent, there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158022.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158025.png" /> has at least one zero component; iii) any right ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158026.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158027.png" /> right linearly dependent elements has fewer than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158028.png" /> generators; and iv) any right ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158029.png" /> on at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158030.png" /> generators is free of unique rank. These properties are also equivalent to their left-right analogues. There are several more equivalent conditions, cf. [[#References|[a1]]].
+
Consider dependence relations of the form $  x \cdot y = x _ {1} y _ {1} + {} \dots + x _ {n} y _ {n} = 0 $,
 +
$  x _ {i} , y _ {i} \in R $(
 +
$  x $
 +
a row vector, $  y $
 +
a column vector). Such a relation is called trivial if for each $  i = 1 \dots n $
 +
either $  x _ {i} = 0 $
 +
or $  y _ {i} = 0 $.  
 +
An $  n $-
 +
term relation $  x \cdot y = 0 $
 +
is trivialized by an invertible $  n \times n $
 +
matrix $  M $
 +
if the relation $  ( xM) ( M  ^ {-} 1 y) $
 +
is trivial. Now let $  R $
 +
be a non-zero ring with unit element, then the following properties are all equivalent: i) every $  m $-
 +
term relation $  \sum x _ {i} y _ {i} = 0 $,  
 +
$  m \leq  n $,  
 +
can be trivialized by an invertible $  m \times m $
 +
matrix; ii) given $  x _ {1} \dots x _ {n} \in R $,  
 +
$  m \leq  n $,  
 +
which are right linearly dependent, there exist $  ( m \times m ) $-
 +
matrices $  M , N $
 +
such that $  MN = I _ {m} $
 +
and $  ( x _ {1} \dots x _ {m} ) M $
 +
has at least one zero component; iii) any right ideal of $  R $
 +
generated by $  m \leq  n $
 +
right linearly dependent elements has fewer than $  m $
 +
generators; and iv) any right ideal of $  R $
 +
on at most $  n $
 +
generators is free of unique rank. These properties are also equivalent to their left-right analogues. There are several more equivalent conditions, cf. [[#References|[a1]]].
  
A ring which satisfies these conditions is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158032.png" />-fir. A ring which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158033.png" />-fir for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158034.png" /> is called a semi-fir.
+
A ring which satisfies these conditions is called an $  n $-
 +
fir. A ring which is an $  n $-
 +
fir for all $  n $
 +
is called a semi-fir.
  
An integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158035.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158037.png" /> (the Ore condition) is called a right Ore ring (cf. also [[Associative rings and algebras|Associative rings and algebras]] for Ore's theorem). It follows that a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158038.png" /> is a Bezout domain (cf. [[Bezout ring|Bezout ring]]) if and only if it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158039.png" />-fir and a right Ore ring.
+
An integral domain $  R $
 +
satisfying $  aR \cap bR \neq \{ 0 \} $
 +
for all $  a , b \in R  ^ {*} = R \setminus  \{ 0 \} $(
 +
the Ore condition) is called a right Ore ring (cf. also [[Associative rings and algebras|Associative rings and algebras]] for Ore's theorem). It follows that a ring $  R $
 +
is a Bezout domain (cf. [[Bezout ring|Bezout ring]]) if and only if it is a $  2 $-
 +
fir and a right Ore ring.
  
For any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158040.png" /> the following are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158041.png" /> is a total matrix ring over a semi-fir; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158042.png" /> is Morita equivalent (cf. [[Morita equivalence|Morita equivalence]]) to a semi-fir; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158043.png" /> is right semi-hereditary (i.e. all finitely-generated right ideals are projective) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158044.png" /> is projective-trivial; and 4) the left-right analogue of 3). Here a ring is projective-trivial if there exists a projective right module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158045.png" /> (called the minimal projective of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158046.png" />) such that every finitely-projective right module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158047.png" /> is the direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158048.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158049.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158050.png" /> unique determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158051.png" />.
+
For any ring $  R $
 +
the following are equivalent: 1) $  R $
 +
is a total matrix ring over a semi-fir; 2) $  R $
 +
is Morita equivalent (cf. [[Morita equivalence|Morita equivalence]]) to a semi-fir; 3) $  R $
 +
is right semi-hereditary (i.e. all finitely-generated right ideals are projective) and $  R $
 +
is projective-trivial; and 4) the left-right analogue of 3). Here a ring is projective-trivial if there exists a projective right module $  P $(
 +
called the minimal projective of $  R $)  
 +
such that every finitely-projective right module $  M $
 +
is the direct sum of $  n $
 +
copies of $  P $
 +
for some $  n $
 +
unique determined by $  M $.
  
For any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158052.png" /> the following are equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158053.png" /> is a total matrix ring over a right fir; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158054.png" /> is Morita equivalent to a right fir; and c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158055.png" /> is right hereditary (i.e. all right ideals are projective) and projective-trivial.
+
For any ring $  R $
 +
the following are equivalent: a) $  R $
 +
is a total matrix ring over a right fir; b) $  R $
 +
is Morita equivalent to a right fir; and c) $  R $
 +
is right hereditary (i.e. all right ideals are projective) and projective-trivial.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158056.png" /> is a semi-fir, then a right module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158057.png" /> is flat if and only if every finitely-generated submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158058.png" /> is free (i.e. if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041580/f04158059.png" /> is locally free).
+
If $  R $
 +
is a semi-fir, then a right module $  P $
 +
is flat if and only if every finitely-generated submodule of $  P $
 +
is free (i.e. if and only if $  P $
 +
is locally free).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1971)</TD></TR></table>

Revision as of 19:40, 5 June 2020


fir.

A (non-commutative) ring (with unit element) in which all (one-sided) ideals are free. More precisely, a right fir is a ring $ R $ in which all right ideals are free of unique rank, as right $ R $- modules. A left fir is defined correspondingly. Firs may be regarded as generalizing the notion of a principal ideal domain.

Consider dependence relations of the form $ x \cdot y = x _ {1} y _ {1} + {} \dots + x _ {n} y _ {n} = 0 $, $ x _ {i} , y _ {i} \in R $( $ x $ a row vector, $ y $ a column vector). Such a relation is called trivial if for each $ i = 1 \dots n $ either $ x _ {i} = 0 $ or $ y _ {i} = 0 $. An $ n $- term relation $ x \cdot y = 0 $ is trivialized by an invertible $ n \times n $ matrix $ M $ if the relation $ ( xM) ( M ^ {-} 1 y) $ is trivial. Now let $ R $ be a non-zero ring with unit element, then the following properties are all equivalent: i) every $ m $- term relation $ \sum x _ {i} y _ {i} = 0 $, $ m \leq n $, can be trivialized by an invertible $ m \times m $ matrix; ii) given $ x _ {1} \dots x _ {n} \in R $, $ m \leq n $, which are right linearly dependent, there exist $ ( m \times m ) $- matrices $ M , N $ such that $ MN = I _ {m} $ and $ ( x _ {1} \dots x _ {m} ) M $ has at least one zero component; iii) any right ideal of $ R $ generated by $ m \leq n $ right linearly dependent elements has fewer than $ m $ generators; and iv) any right ideal of $ R $ on at most $ n $ generators is free of unique rank. These properties are also equivalent to their left-right analogues. There are several more equivalent conditions, cf. [a1].

A ring which satisfies these conditions is called an $ n $- fir. A ring which is an $ n $- fir for all $ n $ is called a semi-fir.

An integral domain $ R $ satisfying $ aR \cap bR \neq \{ 0 \} $ for all $ a , b \in R ^ {*} = R \setminus \{ 0 \} $( the Ore condition) is called a right Ore ring (cf. also Associative rings and algebras for Ore's theorem). It follows that a ring $ R $ is a Bezout domain (cf. Bezout ring) if and only if it is a $ 2 $- fir and a right Ore ring.

For any ring $ R $ the following are equivalent: 1) $ R $ is a total matrix ring over a semi-fir; 2) $ R $ is Morita equivalent (cf. Morita equivalence) to a semi-fir; 3) $ R $ is right semi-hereditary (i.e. all finitely-generated right ideals are projective) and $ R $ is projective-trivial; and 4) the left-right analogue of 3). Here a ring is projective-trivial if there exists a projective right module $ P $( called the minimal projective of $ R $) such that every finitely-projective right module $ M $ is the direct sum of $ n $ copies of $ P $ for some $ n $ unique determined by $ M $.

For any ring $ R $ the following are equivalent: a) $ R $ is a total matrix ring over a right fir; b) $ R $ is Morita equivalent to a right fir; and c) $ R $ is right hereditary (i.e. all right ideals are projective) and projective-trivial.

If $ R $ is a semi-fir, then a right module $ P $ is flat if and only if every finitely-generated submodule of $ P $ is free (i.e. if and only if $ P $ is locally free).

References

[a1] P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
How to Cite This Entry:
Free ideal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_ideal_ring&oldid=14580