Characteristic Direct product of the Rings

Question

Let the rings, $R_i$ and its direct product, $R = \Pi_1 ^{n}R_i = R_1 \times R_2 \times ... \times R_n$

Say the $Char(R_i) = m_i$

(1) $Char(R)$ = $lcm(m_1,m_2,...m_n)$

If all the rings, $R_i$ are commutative, The statement (1) is surely true.

But what if the There are some rings that not commutative.

Does statement (1) is true?

(I.E. IS still $(1)$ true that regardelss of the $R_i$ is a commutative or not ?)

Thanks.

Answer 1

Generally, if you have two groups $G_1$ and $G_2$, then the order of an element $ (g_1, g_2) \in G_1 \times G_2$ is simply lcm(ord($g_1$), ord($g_2$)). If the order of one of the elements is infinite, we simply take the lcm to mean infinite.

The underlying additive group of any ring is abelian. And the characteristic of a nonzero unital ring is just the order of the multiplicative identity $1$ in that abelian group.