Even functions absorb composition?

If $f(x)$ and $g(x)$ are real functions and $g$ is even, so is $f(g(x))$. Even functions are also closed under addition. I noticed that these are similar properties to those of an ideal of a ring, except composition doesn't distribute over addition in general so the ring structure isn't there. In abstract algebra are rings without the distributive property studied? If so, are homomorphisms related to sets closed under addition that absorb the multiplication operation, like they are for normal rings?