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单词 Ideal1
释义

ideal


Let S be a semigroup. An ideal of S is a non-empty subset of S which is closed under multiplication on either side by elements of S. Formally, I is an ideal of S if I is non-empty, and for all xI and sS, we have sxI and xsI.

One-sided ideals are defined similarly. A non-empty subset A of S is a left idealMathworldPlanetmathPlanetmath (resp. right ideal) of S if for all aA and sS, we have saA (resp. asA).

A principal left ideal of S is a left ideal generated by a single element. If aS, then the principal left ideal of S generated by a is S1a=Sa{a}. (The notation S1 is explained here (http://planetmath.org/AdjoiningAnIdentityToASemigroup3).)

Similarly, the principal right ideal generated by a is aS1=aS{a}.

The notation L(a) and R(a) are also common for the principal left and right ideals generated by a respectively.

A principal idealMathworldPlanetmathPlanetmath of S is an ideal generated by a single element. The ideal generated by a is

S1aS1=SaSSaaS{a}.

The notation J(a)=S1aS1 is also common.

Titleideal
Canonical nameIdeal1
Date of creation2013-03-22 13:05:43
Last modified on2013-03-22 13:05:43
Ownermclase (549)
Last modified bymclase (549)
Numerical id8
Authormclase (549)
Entry typeDefinition
Classificationmsc 20M12
Classificationmsc 20M10
Related topicReesFactor
Definesleft ideal
Definesright ideal
Definesprincipal ideal
Definesprincipal left ideal
Definesprincipal right ideal
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