# Characteristic Direct product of the Rings

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.

• Addition on a direct product occurs componentwise, and the characteristic of a ring relates to addition of the identity element 1, not to multiplication, so I don't see how commutativity should change anything. Your statement (1) will hold for any finite product of unital rings. Commented Jul 18, 2019 at 3:28

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.
• Remark: The zero ring has characteristic $1$. Commented Jul 18, 2019 at 4:02