isomorphism swapping zero and unity
Let be a ring with unity 1. Define two new binary operations of as follows:
(1) |
Then we see that
(2) |
But moreover, the algebraic system is a unitary ring, too, and isomorphic with the original ring.
In fact, we may define the bijective mapping
(3) |
from to and verify that it is homomorphic:
Thus as a homomorphic image (http://planetmath.org/HomomorphicImageOfGroup) of the ring is a ring, it’s a question of two isomorphic rings.