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An associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747401.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747403.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747404.png" />. Examples of principal ideal rings include the ring of integers, the ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747405.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747406.png" />, the ring of skew polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747407.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747408.png" /> with an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p0747409.png" /> (the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474010.png" /> have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474012.png" />, the addition of these elements is as usual, while their multiplication is defined by the associativity and distributivity laws and by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474013.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474014.png" />), the ring of differential polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474015.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474016.png" /> with a derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474017.png" /> (this ring also consists of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474019.png" />; addition is carried out in the ordinary way while multiplication is determined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474021.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474022.png" /> is a principal ideal domain, then two non-zero elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474025.png" /> have a greatest common left divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474026.png" /> and a least common right multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474027.png" />, which are defined as the elements that satisfy the equations:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474028.png" /></td> </tr></table>
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The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474030.png" /> 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).
+
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:
  
A submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474031.png" /> of a free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474032.png" /> of finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474033.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474034.png" /> is a free module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474035.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474036.png" />, and in the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474038.png" /> it is possible to select bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474040.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474044.png" /> is a complete divisor, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474045.png" />, of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474046.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474047.png" />. Each finitely-generated module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474048.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474049.png" /> is a direct sum of cyclic modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474053.png" /> is a complete divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474054.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474056.png" />. This theorem generalizes the fundamental theorem on finitely-generated Abelian groups (cf. [[Abelian group|Abelian group]]). The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474058.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474059.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474060.png" /> can be represented as a direct sum of indecomposable cyclic modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474063.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474065.png" />, are defined up to a similarity, and are called elementary divisors of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474066.png" />. If the principal ideal domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474067.png" /> is commutative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474068.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474071.png" /> are irreducible (prime) elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474072.png" />. The ordinary properties of elementary divisors and invariant factors of linear transformations of finite-dimensional vector spaces follow from the above statements [[#References|[3]]].
+
$$
 +
aR + bR  = ( a, b) R; \  aR \cap bR  = [ a, b] R.
 +
$$
 +
 
 +
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====
 
====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>
 
<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====
 
====Comments====
The two examples of skew and differential polynomial rings are a special case of the general-skew-polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474074.png" /> is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474076.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474078.png" />-derivation (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474079.png" />), with multiplication defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474080.png" />. This ring is a principal ideal ring. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474081.png" /> is assumed to be only an isomorphism, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074740/p07474082.png" />, then the ring is right principal but not left principal.
+
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.
 
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.

Revision as of 08:07, 6 June 2020


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.

How to Cite This Entry:
Principal ideal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_ideal_ring&oldid=14409
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article