$\#(A \times B) = lcm(\#A , \#B)$ where $A$ and $B$ are unitary and commutative rings?

Question

Is it true that $\#(A \times B) = lcm(\#A , \#B)$ where $A$ and $B$ are unitary and commutative rings? the symbol $\#$ denotes the cardinal. I'm not sure if this statement is true or false, I have not found any counterexample yet. Any help would be appreciated.

Answer 1

This is not true. Consider $A = B = \mathbb{Z} / (2)$. Then $\#A = \#B = 2$, so $lcm(\#A, \#B) = 2$. But $\#(A \times B) = 4$.

In general, we have $\#(A \times B) = (\#A) \times (\#B)$ for any sets $A$, $B$. And obviously it's not always true that $a \times b = lcm(a, b)$.