Principal ideal ring

An associative ring with a unit element (cf. Associative rings and algebras) in which all right and left ideals are principal, i.e. have the form and , respectively, where . Examples of principal ideal rings include the ring of integers, the ring of polynomials over a field , the ring of skew polynomials over a field with an automorphism (the elements of have the form , , the addition of these elements is as usual, while their multiplication is defined by the associativity and distributivity laws and by the equation where ), the ring of differential polynomials over a field with a derivation (this ring also consists of the elements , ; addition is carried out in the ordinary way while multiplication is determined by the equation , ). A principal ideal ring without a zero divisor is called a principal ideal domain. A commutative principal ideal ring is a direct sum of principal ideal domains and a principal ideal ring with a unique nilpotent prime ideal (cf. Nilpotent ideal; Prime ideal). If is a principal ideal domain, then two non-zero elements and of have a greatest common left divisor and a least common right multiple , which are defined as the elements that satisfy the equations:

The elements and are unique, up to an invertible right factor. A principal ideal domain is a unique factorization domain. The two-sided ideals of a principal ideal domain form a free commutative multiplicative semi-group with a zero and a unit element (the maximal ideals of the ring are the free generators of this semi-group).

A submodule of a free module of finite rank over is a free module of rank over , and in the modules and it is possible to select bases and so that , , where and is a complete divisor, i.e. , of the elements if . Each finitely-generated module over is a direct sum of cyclic modules , , where and is a complete divisor of if , . This theorem generalizes the fundamental theorem on finitely-generated Abelian groups (cf. Abelian group). The elements , , in the preceding theorem are unambiguously defined up to a similarity (cf. Associative rings and algebras). These elements are called invariant factors of . Moreover, can be represented as a direct sum of indecomposable cyclic modules , where , . The elements , , are defined up to a similarity, and are called elementary divisors of the module . If the principal ideal domain is commutative, then or , , where are irreducible (prime) elements of . The ordinary properties of elementary divisors and invariant factors of linear transformations of finite-dimensional vector spaces follow from the above statements [3].

References

[1]	N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943)
[2]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[3]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)

Comments

The two examples of skew and differential polynomial rings are a special case of the general-skew-polynomial ring , where is an automorphism of and is an -derivation (i.e. ), with multiplication defined by . This ring is a principal ideal ring. If is assumed to be only an isomorphism, with , then the ring is right principal but not left principal.

Left (and right) ideals of rings of finite matrices which contain a non-zero divisor matrix are also left (right) principal. The module properties, mentioned above, have also a (the original) version for matrices: i.e. every matrix over these rings is equivalent to a matrix in diagonal form.