mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are

free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...

s with unique

rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...

. A ring such that all right ideals with at most ''n'' generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules 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

domain A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...

. Furthermore, a

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

fir is precisely a

principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...

, while a commutative semifir is precisely a

Bézout domain In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is also a principal ideal. This means that Bézout's identity holds for every pair of elements, and that every finitely generated ideal is principal. ...

. 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

is a semifir. Since all right ideals of a right fir are free, they are projective. So, any right fir is a right

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. If this is required only for finitely generated submodul ...

, and likewise a right semifir is a right

semihereditary 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. If this is required only for finitely generated submodul ...

. Because

projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizati ...

s over

local ring In mathematics, more specifically 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 algebraic varieties or manifolds, or of ...

s are free, and because local rings have

invariant basis number In the mathematical field of ring theory, a ring ''R'' has the invariant basis number (IBN) property if all finitely generated free modules over ''R'' have a well-defined rank. In the case of fields, the IBN property is the fact that finite-dime ...

, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir. Unlike a principal right ideal domain, a right fir is not necessarily right

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...

, however in the commutative case, ''R'' is a

Dedekind domain In mathematics, 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 un ...

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 ring In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

s . Semifirs have

and every semifir is a

Sylvester domain Sylvester or Silvester is a name derived from the Latin adjective ''silvestris'' meaning "wooded" or "wild", which derives from the noun ''silva'' meaning "woodland". Classical Latin spells this with ''i''. In Classical Latin, ''y'' represented a ...

References

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Properties and examples

References

Further reading