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单词 IsomorphismSwappingZeroAndUnity
释义

isomorphism swapping zero and unity


Let  (R,+,)  be a ring with unity 1.  Define two new binary operationsMathworldPlanetmath of R as follows:

ab=:a+b-1,ab=:a+b-ab(1)

Then we see that

a1=a= 1a,a0=a= 0a.(2)

But moreover, the algebraic system  (R,,)  is a unitary ring, too, and isomorphicPlanetmathPlanetmathPlanetmath with the original ring.

In fact, we may define the bijectiveMathworldPlanetmathPlanetmath mapping

f:x 1-x(3)

from R to R and verify that it is homomorphic:

f(a)f(b)=(1-a)(1-b)=(1-a)+(1-b)-1= 1-a-b=f(a+b),
f(a)f(b)=(1-a)(1-b)=(1-a)+(1-b)-(1-a)(1-b)= 1-ab=f(ab)

Thus  (R,,)  as a homomorphic imagePlanetmathPlanetmathPlanetmath (http://planetmath.org/HomomorphicImageOfGroup) of the ring  (R,+,)  is a ring, it’s a question of two isomorphic rings.

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更新时间:2025/5/4 18:39:26