# 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 .