Is the characteristic of ring invariant under homomorphism?

Question

I'm completing my notes on the characteristic of a ring and I find these question:

Under which conditions, can we guarantee that two rings $A,B$ are not homomorphic given their characteristics?

So I wonder, is the characteristic invariant under homormorphism?

Answer 1

The characteristic is not invariant, consider $\Bbb Z_6\to \Bbb Z_6/(3)\cong \Bbb Z_3$, which is a homomorphism form a ring with characteristic $6$ to a ring with characteristic $3$.

However, that doesn't mean all hope is lost for the more general question: Let $f:A\to B$ be a homomorphism. By the definition of homomorphism, we must have $$ 0 = f(0) = f(1_A\cdot \operatorname{char} A) = 1_B\cdot \operatorname{char} A $$ so we must have $\operatorname{char}B \mid \operatorname{char} A$ (note that this argument and especially the result also applies if either characteristic is $0$). So if the characteristic of $B$ does not divide the characteristic of $A$, then there is no homomorphism $A\to B$.