Homomorphism between a ring which is a boolean algebra and one which is not.

Question

I remember reading in a textbook that there can exist a homomorphism between a ring which is a boolean algebra and one which is not. Can anyone give me some example of this.

Answer 1

Let $R$ be any (nontrivial) Boolean ring (such as $\mathbb Z_2$). Then $R[X]$ is not a Boolean ring (because $X^2\ne X$).

The injection $R\to R[X]$ is a homomorphism, and so is the evaluation map $R[X] \to R$ sending $X\mapsto 0$.

If you want finite examples, choose $R$ to be finite and consider the quotient $R[X]/\langle X^2\rangle$ instead of $R[X]$.