A ring homomorphism $f:R \to S$ induce a ring homomorphism $H^*(X,A;R)\to H^*(X,A;S)$ on cohomology rings

Question

I know that for a group homomorphism $f:G \to H$ and a pair of spaces $(X,A)$, $f$ induces a group homomorphism on homology $f_*:H_n(X,A;G) \to H_n(X,A,H).$

Similarly, does a ring homomorphism $f:R \to S$ induce a ring homomorphism $H^*(X,A;R)\to H^*(X,A;S)$ on cohomology rings?

Answer 1

Yes, it does. You simply take any cohomology class and compose with the ring homomorphism $f$. The only thing requiring checking is that the induced map is then a ring homomorphism, and it is.