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 in which all right ideals are free of unique rank, as right -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 , ( a row vector, a column vector). Such a relation is called trivial if for each either or . An -term relation is trivialized by an invertible matrix if the relation is trivial. Now let be a non-zero ring with unit element, then the following properties are all equivalent: i) every -term relation , , can be trivialized by an invertible matrix; ii) given , , which are right linearly dependent, there exist -matrices such that and has at least one zero component; iii) any right ideal of generated by right linearly dependent elements has fewer than generators; and iv) any right ideal of on at most 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 -fir. A ring which is an -fir for all is called a semi-fir.

An integral domain satisfying for all (the Ore condition) is called a right Ore ring (cf. also Associative rings and algebras for Ore's theorem). It follows that a ring is a Bezout domain (cf. Bezout ring) if and only if it is a -fir and a right Ore ring.

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

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

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

References

[a1]	P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)