Prove that $f(R) = \{ y \in P\ | \; (\exists x \in R)\; f(x) = y \; \} $ is a subring of ring $P$.

Question

Let $f$ be a homomorphism of ring $R$ into ring $P$. Prove that $f(R) = \{ y \in P\ | \; (\exists x \in R)\; f(x) = y \; \} $ is a subring of ring $P$.

What would be a good way to prove this? I know how to do subring test, but I don't know how to use it here.

Answer 1

To be really annoying, you need to check that $0_P\in f(R)$, $1_P\in f(R)$ if they are unital, and that if $f(r),f(s)\in f(R)$ then $f(r)-f(s)\in f(R)$ and $f(r)\cdot f(s)\in f(R)$. Essentially, you have to prove that $f(R)$ is a subgroup with respect to $+$ and a subsemigroup (or submonoid, if they are unital) with respect to $\cdot$. Now, try to use the definition of ring homomorphism.