Principal ideal ring

An associative ring $ R $ with a unit element (cf. Associative rings and algebras) in which all right and left ideals are principal, i.e. have the form $ aR $ and $ Ra $, respectively, where $ a \in R $. Examples of principal ideal rings include the ring of integers, the ring of polynomials $ F ( x) $ over a field $ F $, the ring of skew polynomials $ F( x, S) $ over a field $ F $ with an automorphism $ S: F \rightarrow F $( the elements of $ F( x, S) $ have the form $\sum_{i=0}^ {n} x ^ {i} a _ {i} $, $ a _ {i} \in F $, the addition of these elements is as usual, while their multiplication is defined by the associativity and distributivity laws and by the equation $ ax = xa ^ {S} $ where $ a \in F $), the ring of differential polynomials $ F( x, \prime ) $ over a field $ F $ with a derivation $ {} \prime : F \rightarrow F $( this ring also consists of the elements $\sum_{i=0}^ {n} x ^ {i} a _ {i} $, $ a _ {i} \in F $; addition is carried out in the ordinary way while multiplication is determined by the equation $ ax = xa + a \prime $, $ a \in F $). 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 $ R $ is a principal ideal domain, then two non-zero elements $ a $ and $ b $ of $ R $ have a greatest common left divisor $ ( a, b) $ and a least common right multiple $ [ a, b] $, which are defined as the elements that satisfy the equations:

$$ aR + bR = ( a, b) R; \ aR \cap bR = [ a, b] R. $$

The elements $ ( a, b) $ and $ [ a, b] $ 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 $ N $ of a free module $ M $ of finite rank $ n $ over $ R $ is a free module of rank $ k \leq n $ over $ R $, and in the modules $ M $ and $ N $ it is possible to select bases $ a _ {1} \dots a _ {n} $ and $ b _ {1} \dots b _ {k} $ so that $ b _ {i} = e _ {i} a _ {i} $, $ 1 \leq i \leq k $, where $ e _ {i} \in R $ and $ e _ {i} $ is a complete divisor, i.e. $ e _ {i} R \cap R e _ {i} \supseteq R e _ {j} R $, of the elements $ e _ {j} $ if $ i < j $. Each finitely-generated module $ K $ over $ R $ is a direct sum of cyclic modules $ R/e _ {i} R $, $ 1 \leq i \leq m $, where $ e _ {i} \in R $ and $ e _ {i} $ is a complete divisor of $ e _ {j} $ if $ i < j $, $ e _ {i} \neq 0 $. This theorem generalizes the fundamental theorem on finitely-generated Abelian groups (cf. Abelian group). The elements $ e _ {i} $, $ 1 \leq i \leq m $, in the preceding theorem are unambiguously defined up to a similarity (cf. Associative rings and algebras). These elements are called invariant factors of $ K $. Moreover, $ K $ can be represented as a direct sum of indecomposable cyclic modules $ R/ q _ {i} R $, where $ q _ {i} \in R $, $ 1 \leq i \leq k $. The elements $ q _ {i} $, $ 1 \leq i \leq k $, are defined up to a similarity, and are called elementary divisors of the module $ K $. If the principal ideal domain $ R $ is commutative, then $ q _ {i} R = 0 $ or $ q _ {i} R = p _ {i} ^ {n _ {i} } R $, $ 1 \leq i \leq k $, where $ p _ {i} $ are irreducible (prime) elements of $ R $. 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 $ F [ x; S, d] $, where $ S $ is an automorphism of $ F $ and $ d $ is an $ S $- derivation (i.e. $ d( ab) = a ^ {S} d( b)+ d( a) b $), with multiplication defined by $ ax = xa ^ {S} + d( a) $. This ring is a principal ideal ring. If $ S $ is assumed to be only an isomorphism, with $ F ^ { S } \neq F $, 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.