The dimension of a ring is defined as the length of a longest prime chain as usual.
Let $A,B$ be affine rings over a field $k$. Then $$\dim A\otimes_k B = \dim A + \dim B.$$ How can we prove or disprove this?
I think the easiest way to prove this uses Noether normalization. [See this handout of Hochster’s.] To give more of a hint, you know that there is a polynomial subalgebra $k[T_1, \dots, T_n] \subseteq A$ over which $A$ is finite, where $n = \dim A$. Similarly, say $B$ is finite over $k[S_1, \dots, S_m]$. Can you write down an $(n + m)$-variable polynomial subalgebra of $A \otimes_k B$? Is $A \otimes_k B$ finite over this subring? Why is that enough?