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单词 Unity
释义

unity


The unity of a ring  (R,+,)  is the multiplicative identityPlanetmathPlanetmath of the ring, if it has such.  The unity is often denoted by e, u or 1.  Thus, the unity satisfies

ea=ae=aaR.

If R consists of the mere 0, then 0 is its unity, since in every ring,  0a=a0=0.  Conversely, if 0 is the unity in some ring R, then  R={0}  (because  a=0a=0aR).

Note.  When considering a ring R it is often mentioned that “…having 10” or that “…with non-zero unity”, sometimes only “…with unity” or “…with ”; all these exclude the case  R={0}.

Theorem.

An element u of a ring R is the unity iff u is an idempotentPlanetmathPlanetmath and regular elementPlanetmathPlanetmath.

Proof.  Let u be an idempotent and regular element.  For any element x of R we have

ux=u2x=u(ux),

and because u is no left zero divisor, it may be cancelled from the equation; thus we get  x=ux.  Similarly,  x=xu.  So u is the unity of the ring.  The other half of the is apparent.

Titleunity
Canonical nameUnity
Date of creation2013-03-22 14:47:17
Last modified on2013-03-22 14:47:17
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id15
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 20-00
Classificationmsc 16-00
Classificationmsc 13-00
Synonymmultiplicative identity
Synonymcharacterization of unity
Related topicZeroDivisor
Related topicRootOfUnity
Related topicZeroRing
Related topicNonZeroDivisorsOfFiniteRing
Related topicOppositePolynomial
Definesnon-zero unity
Definesnonzero unity
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更新时间:2025/5/4 22:45:59