Characteristic of Integral-domain where $15a=0$ but $3b\neq 0$.

Question

Let $R$ be an integral domain. Let $a,b \in R$. Assume that a and b both not zeros, $15a = 0$ and $3b \neq 0$ group. What can you say about the characteristic of $R$?

Answer 1

$0 = 15a = 5a\cdot 3a$, and $R$ is an integral domain, so $3a=0$ or $5a=0$. Now use the integral domain property again.

Can you now deduce what the characteristic is? (Recall, it cannot be composite and we know $3b \neq 0$).

Answer 2

Hint $\,\ 0 = (15a)b = 5(3b)a\ $ and $\ (3b)a\ne 0\,\Rightarrow\,\ldots = 0$