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Difference between revisions of "Principal ideal"

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$$n\alpha+t\alpha+\alpha s+\sum _{ i }{ [(k_{ i }\alpha)l_{ i }+k_{ i }^\prime (\alpha l_{ i }^\prime )] } $$
 
$$n\alpha+t\alpha+\alpha s+\sum _{ i }{ [(k_{ i }\alpha)l_{ i }+k_{ i }^\prime (\alpha l_{ i }^\prime )] } $$
  
where $k,t,s,k_{i},k_{i}^{\prime},l_{i},l_{i}^{\prime}$ are arbitrary elements of $K$ and $n\alpha=\alpha+\dots +\alpha$ $(n\quad terms,\quad n\in\Zeta )$. If $K$ is a ring with a unit element, the term $n\alpha$ may be omitted. In particular, for an algebra $A$ over a field,
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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, $n\in\Zeta$). If $K$ is a ring with a unit element, the term $n\alpha$ may be omitted. In particular, for an algebra $A$ over a field,
  
 
$$L(a)=A\alpha,\qquad R(a)=\alpha A,\qquad J(\alpha)=A\alpha A.$$
 
$$L(a)=A\alpha,\qquad R(a)=\alpha A,\qquad J(\alpha)=A\alpha A.$$
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====References====
 
====References====
<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>
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|valign="top"|{{Ref|Be}}||valign="top"| L. Beran,  "Orthomodular lattices", Reidel  (1985)  pp. 4ff
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|valign="top"|{{Ref|Gr}}||valign="top"| G. Grätzer,  "Lattice theory", Freeman  (1971)
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|valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh,  "Lectures on general algebra", Chelsea  (1963)  pp. 78; 86; 162  (Translated from Russian)
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|valign="top"|{{Ref|Ly}}||valign="top"|  E.S. Lyapin,  "Semigroups", Amer. Math. Soc.  (1963)  pp. Chapt. IV, Sect. 3  (Translated from Russian)
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Revision as of 18:18, 20 April 2012

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

The left principal ideal $L(\alpha)$ of a ring $K$ 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, $n\in\Zeta$). If $K$ is a ring with a unit element, the term $n\alpha$ may be omitted. In particular, for an algebra $A$ over a field,

$$L(a)=A\alpha,\qquad R(a)=\alpha A,\qquad J(\alpha)=A\alpha A.$$

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.

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 $A$ 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 $r\in K$.

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=24901
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