Find the characteristic of the ring $\mathbb Z_6 \times \mathbb Z_{15}$

Question

My attempt: Let the characteristic be $n$.

Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$,

i.e. $n \cdot 1_6=0_6$ and $n \cdot 1_{15}=0_{15}$

The least $n$ for which both are true is $30$, so $30$ is the characteristic.

Is my method correct? If so, if my writing ok?

Answer 1

Yes, in general the characteristic of

$$\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \cdots \times \mathbb{Z}_{n_k}$$

is $LCM(n_1,..,n_k$).

Answer 2

yes, you have to calculate lcm of $6$ and $15$.

Answer 3

$\mathbb Z_6 \times \mathbb Z_{15}$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_{30}$, which makes it clear that the characteristic is $30$.