Free ideal ring
Encyclopedia
In mathematics
, especially in the field of ring theory
, a (left) free ideal ring, or fir, is a ring in which all left ideals
are free
of unique rank. A ring such that all left ideals with at most n generators is free of unique rank is called an n-fir. A semifir
is a ring in which all finitely generated left ideals are free of unique rank. (Thus, a ring is semifir if it is n-fir for all n ≥ 0.) The semifir property is left-right symmetric, but the fir property is not.
. Furthermore, a commutative
fir is precisely a principal ideal domain
, while a commutative semifir is precisely a Bézout domain
. These last facts are not generally true for noncommutative rings, however .
Every principal right ideal domain
R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R. In the same way, a right Bézout domain
is a semifir.
Since all right ideals of a right fir are free, they are projective, hence any right fir is a hereditary ring
, and likewise a semifir is a right and left semihereditary ring. As a partial converse, since it is known that projective module
s over local ring
s are free, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a semifir.
Unlike a principal right idea domain, a right fir is not necessarily right Noetherian
, however in the commutative case, R is a Dedekind domain
since it is a hereditary domain, and so is necessarily Noetherian.
Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the field of ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, a (left) free ideal ring, or fir, is a ring in which all left ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
are free
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
of unique rank. A ring such that all left ideals with at most n generators is free of unique rank is called an n-fir. A semifir
Semifir
In algebra, a semifir is a ring in which every finitely generated right ideal is free of unique rank. Similarly, a fir is a ring in which every right ideal is free of unique rank. Every fir is thus a semifir. A commutative semifir is a Bézout domain, while a commutative fir is a principal ideal...
is a ring in which all finitely generated left ideals are free of unique rank. (Thus, a ring is semifir if it is n-fir for all n ≥ 0.) The semifir property is left-right symmetric, but the fir property is not.
Properties and examples
It turns out that a left and right fir is a domainDomain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
. Furthermore, a commutative
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
fir is precisely a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, while a commutative semifir is precisely a Bézout domain
Bézout domain
In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal...
. These last facts are not generally true for noncommutative rings, however .
Every principal right ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R. In the same way, a right Bézout domain
Bézout domain
In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal...
is a semifir.
Since all right ideals of a right fir are free, they are projective, hence any right fir is a hereditary ring
Hereditary ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective...
, and likewise a semifir is a right and left semihereditary ring. As a partial converse, since it is known that projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s over local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
s are free, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a semifir.
Unlike a principal right idea domain, a right fir is not necessarily right Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
, however in the commutative case, R is a Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
since it is a hereditary domain, and so is necessarily Noetherian.
Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings .