Namespaces
Variants
Actions

Difference between revisions of "Principal ideal"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (links)
 
(8 intermediate revisions by 4 users not shown)
Line 1: Line 1:
An [[Ideal|ideal]] (of a ring, algebra, semi-group, or lattice) generated by one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747301.png" />, i.e. the smallest ideal containing the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747302.png" />.
+
{{TEX|done}}
  
The left principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747303.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747304.png" /> contains, in addition to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747305.png" /> itself, also all the elements
+
An [[ideal]] (of a [[ring]], algebra, [[semi-group]], or [[lattice]]) generated by one element $\alpha$, i.e. the smallest ideal $L(\alpha)$ containing the element $\alpha$.
  
<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/p074730/p0747306.png" /></td> </tr></table>
+
===Rings===
 +
The left principal ideal $L(\alpha)$ of a ring $A$ contains, in addition to the element $\alpha$ itself, also all the elements
  
the right principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747307.png" /> contains all the elements
+
$$k\alpha+n\alpha$$
  
<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/p074730/p0747308.png" /></td> </tr></table>
+
the right principal ideal $R(\alpha)$ contains all the elements
  
and the two-sided principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747309.png" /> contains all elements of the form
+
$$\alpha k+n\alpha$$
  
<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/p074730/p07473010.png" /></td> </tr></table>
+
and the two-sided principal ideal $J(\alpha)$ contains all elements of the form
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473011.png" /> are arbitrary elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473014.png" /> terms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473015.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473016.png" /> is a ring with a unit element, the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473017.png" /> may be omitted. In particular, for an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473018.png" /> over a field,
+
$$n\alpha+t\alpha+\alpha s+\sum _{ i }{ [(k_{ i }\alpha)l_{ i }+k_{ i }^\prime (\alpha l_{ i }^\prime )] } $$
  
<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/p074730/p07473019.png" /></td> </tr></table>
+
where $k,t,s,k_{i},k_{i}^{\prime},l_{i},l_{i}^{\prime}$ are arbitrary elements of $K$ and $n\alpha=\alpha+\dots +\alpha$ ($n$ terms, $n\in\Z$). If $A$ is a ring with a unit element, in particular, for an algebra $A$ over a field, the term $n\alpha$ may be omitted:
 +
$$
 +
L(a)=A\alpha,\qquad R(a)=\alpha A,\qquad J(\alpha)=A\alpha A\ .
 +
$$
  
In a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473020.png" /> one also has left, right and two-sided ideals generated by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473021.png" />, and they are equal, respectively, to
+
====Comments====
 +
Let $A$ be an [[integral domain]] with [[field of fractions]] $K$. A principal fractional ideal of $A$ is an $A$-submodule of $K$ of the form $Ar$ for some $r\in K$.
 +
 
 +
===Semigroups===
 +
In a semi-group $S$ one also has left, right and two-sided ideals generated by an element $\alpha$, and they are equal, respectively, to
  
<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/p074730/p07473022.png" /></td> </tr></table>
+
$$L(\alpha)=S^1\alpha,\qquad R(\alpha)=\alpha S^1,\qquad L(\alpha)=S^1\alpha S^1,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473023.png" /> is the semi-group coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473024.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473025.png" /> contains a unit, and is otherwise obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473026.png" /> by external adjunction of a unit.
+
where $S^1$ is the semi-group coinciding with $S$ if $S$ contains a unit, and is otherwise obtained from $S$ by external adjunction of a unit.
  
The principal ideal of a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473027.png" /> generated by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473028.png" /> is identical with the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473030.png" />; it is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473032.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473033.png" /> if the lattice has a zero. Thus,
+
===Lattices===
 +
The principal ideal of a lattice $L$ generated by an element $\alpha$ is identical with the set of all $x$ such that $x \le \alpha$; it is usually denoted by $\alpha^\Delta$,$[{\alpha}]$, or $[{0,\alpha}]$ if the lattice has a zero. Thus,
  
<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/p074730/p07473034.png" /></td> </tr></table>
+
$$\alpha^{ \Delta  }=\alpha L=\{ { \alpha x:x\in L }\} .$$
  
 
In a lattice of finite length all ideals are principal.
 
In a lattice of finite length all ideals are principal.
Line 32: Line 41:
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473035.png" /> be an integral domain with field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473036.png" />. A principal fractional ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473038.png" />-submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473039.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473040.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473041.png" />.
+
Let $L$ be a lattice. Dual to the principal ideal generated by $\alpha \in L$ one has the principal dual ideal or principal filter determined by $\alpha$, which is the set $[\alpha)=\left\{ y\in L:\alpha \le y \right\} $. The principal ideal in $L$ determined by $\alpha$ is also denoted (more accurately) by $[\alpha)$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473042.png" /> be a lattice. Dual to the principal ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473043.png" /> one has the principal dual ideal or principal filter determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473044.png" />, which is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473045.png" />. The principal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473046.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473047.png" /> is also denoted (more accurately) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473048.png" />.
+
A partially ordered set is a [[Complete lattice|complete lattice]] if and only if it has a zero and every ideal in $L$ is principal.
  
A partially ordered set is a [[Complete lattice|complete lattice]] if and only if it has a zero and every ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473049.png" /> is principal.
+
====References====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Be}}||valign="top"|  L. Beran,  "Orthomodular lattices", Reidel  (1985)  pp. 4ff
 +
|-
 +
|valign="top"|{{Ref|Gr}}||valign="top"|  G. Grätzer,  "Lattice theory", Freeman  (1971)
 +
|-
 +
|valign="top"|{{Ref|Ku}}||valign="top"|  A.G. Kurosh,  "Lectures on general algebra", Chelsea  (1963)  pp. 78; 86; 162  (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|Ly}}||valign="top"|  E.S. Lyapin,  "Semigroups", Amer. Math. Soc.  (1963)  pp. Chapt. IV, Sect. 3  (Translated from Russian)
 +
|-
 +
|}
  
====References====
+
[[Category:General algebraic systems]]
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1963)  pp. Chapt. IV, Sect. 3  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Beran,  "Orthomodular lattices" , Reidel  (1985)  pp. 4ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Grätzer,  "Lattice theory" , Freeman  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  pp. 78; 86; 162  (Translated from Russian)</TD></TR></table>
 

Latest revision as of 20:54, 28 November 2014


An ideal (of a ring, algebra, semi-group, or lattice) generated by one element $\alpha$, i.e. the smallest ideal $L(\alpha)$ containing the element $\alpha$.

Rings

The left principal ideal $L(\alpha)$ of a ring $A$ contains, in addition to the element $\alpha$ itself, also all the elements

$$k\alpha+n\alpha$$

the right principal ideal $R(\alpha)$ contains all the elements

$$\alpha k+n\alpha$$

and the two-sided principal ideal $J(\alpha)$ contains all elements of the form

$$n\alpha+t\alpha+\alpha s+\sum _{ i }{ [(k_{ i }\alpha)l_{ i }+k_{ i }^\prime (\alpha l_{ i }^\prime )] } $$

where $k,t,s,k_{i},k_{i}^{\prime},l_{i},l_{i}^{\prime}$ are arbitrary elements of $K$ and $n\alpha=\alpha+\dots +\alpha$ ($n$ terms, $n\in\Z$). If $A$ is a ring with a unit element, in particular, for an algebra $A$ over a field, the term $n\alpha$ may be omitted: $$ L(a)=A\alpha,\qquad R(a)=\alpha A,\qquad J(\alpha)=A\alpha A\ . $$

Comments

Let $A$ be an integral domain with field of fractions $K$. A principal fractional ideal of $A$ is an $A$-submodule of $K$ of the form $Ar$ for some $r\in K$.

Semigroups

In a semi-group $S$ one also has left, right and two-sided ideals generated by an element $\alpha$, and they are equal, respectively, to

$$L(\alpha)=S^1\alpha,\qquad R(\alpha)=\alpha S^1,\qquad L(\alpha)=S^1\alpha S^1,$$

where $S^1$ is the semi-group coinciding with $S$ if $S$ contains a unit, and is otherwise obtained from $S$ by external adjunction of a unit.

Lattices

The principal ideal of a lattice $L$ generated by an element $\alpha$ is identical with the set of all $x$ such that $x \le \alpha$; it is usually denoted by $\alpha^\Delta$,$[{\alpha}]$, or $[{0,\alpha}]$ if the lattice has a zero. Thus,

$$\alpha^{ \Delta }=\alpha L=\{ { \alpha x:x\in L }\} .$$

In a lattice of finite length all ideals are principal.


Comments

Let $L$ be a lattice. Dual to the principal ideal generated by $\alpha \in L$ one has the principal dual ideal or principal filter determined by $\alpha$, which is the set $[\alpha)=\left\{ y\in L:\alpha \le y \right\} $. The principal ideal in $L$ determined by $\alpha$ is also denoted (more accurately) by $[\alpha)$.

A partially ordered set is a complete lattice if and only if it has a zero and every ideal in $L$ is principal.

References

[Be] L. Beran, "Orthomodular lattices", Reidel (1985) pp. 4ff
[Gr] G. Grätzer, "Lattice theory", Freeman (1971)
[Ku] A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) pp. 78; 86; 162 (Translated from Russian)
[Ly] E.S. Lyapin, "Semigroups", Amer. Math. Soc. (1963) pp. Chapt. IV, Sect. 3 (Translated from Russian)
How to Cite This Entry:
Principal ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_ideal&oldid=12133
This article was adapted from an original article by V.N. RemeslennikovT.S. FofanovaL.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article