Free ideal ring

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)