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An associative ring  $  R $
with a unit element (cf. [[Associative rings and algebras|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|Nilpotent ideal]]; [[Prime 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|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|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|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 [[#References|[3]]].

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

====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.