# When one say two cohomology rings are isomorphic, does the degree of generator matters? [duplicate]

My question is when one say two cohomology rings are isomorphic, does the degree of generator matters?

Here is a motivating example:

$H^*(RP^{\infty}, \mathbb Z_2)\cong \mathbb Z_2[\alpha]$ with $|\alpha|=1$, and $H^*(CP^{\infty}, \mathbb Z_2)\cong \mathbb Z_2[\beta]$, with $|\beta|=2$, but if we define map between two cohomology rings by sending $\alpha$ to $\beta$, it is clearly a ring isomorphism. But we know that the two spaces even don't share the same cohomology groups. So, I have a doubt on definition of isomorphism between cohomology rings.