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单词 Subring
释义

subring


Let (R,+,*) a ring. A subring is a subset S of R with the operationsMathworldPlanetmath + and * of R restricted to S and such that S is a ring by itself.

Notice that the restricted operations inherit the associative and distributive properties of + and *, as well as commutativity of +.So for (S,+,*) to be a ring by itself, we need that (S,+) be a subgroupMathworldPlanetmathPlanetmath of (R,+) and that (S,*) be closed.The subgroup condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to S being non-empty and having the property that x-yS for all x,yS.

A subring S is called a left idealMathworldPlanetmathPlanetmath if for all sS and all rR we have r*sS. Right ideals are defined similarly, with s*r instead of r*s.If S is both a left ideal and a right ideal, then it is called a two-sided ideal. If R is commutativePlanetmathPlanetmathPlanetmath, then all three definitions coincide. In ring theory, ideals are far more important than subrings, as they play a role analogous to normal subgroupsMathworldPlanetmath in group theory.

Example:

Consider the ring (,+,). Then (2,+,) is a subring, since the differencePlanetmathPlanetmath and productPlanetmathPlanetmath of two even numbers is again an even number.

Titlesubring
Canonical nameSubring
Date of creation2013-03-22 12:30:19
Last modified on2013-03-22 12:30:19
Owneryark (2760)
Last modified byyark (2760)
Numerical id17
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20-00
Classificationmsc 16-00
Classificationmsc 13-00
Related topicIdeal
Related topicRing
Related topicGroup
Related topicSubgroup
Definesideal
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更新时间:2025/5/4 6:26:28