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) |
Free ideal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_ideal_ring&oldid=46985