Dimensions of integral ring extensions

Question

If $X$ is a commutative ring with identity and $Y$ is an integral $X$-algebra, show that $\dim\,X=\dim\,Y$.

I think also that $X$ needs to be a subring of $Y$. Why is this true?

Answer 1

If $X$ is not a subring of $Y$, silly things can happen. For example, the map $k[x,y]\to k[x]$ which maps $f(x,y)$ to $f(x,0)$ is surjective, so integral.