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 ( of the ring  (R,+,)  is a ring, it’s a question of two isomorphic rings.

Title isomorphism swapping zero and unity
Canonical name IsomorphismSwappingZeroAndUnity
Date of creation 2013-03-22 19:17:16
Last modified on 2013-03-22 19:17:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 16B99
Classification msc 20A05
Classification msc 16S50
Related topic RingHomomorphism
Related topic EpimorphismBetweenUnitaryRings
Related topic Null
Related topic TranslationAutomorphismOfAPolynomialRing