Prove that the characteristic of the direct sum of $n$ rings is equal to the lcm of the characteristics

Question

Given that $R_{i}$ are rings, I need to prove that

$char(R) = char(R_{1} \bigoplus R_{2} \bigoplus \cdots \bigoplus R_{n}) = lcm(char(R_{1}), char(R_{2}), \cdots, char(R_{n}))$

Recall that the characteristic of a ring $S$, denoted $char(S)$, is the smallest positive integer $n$ such that $nx=0$, $\forall x \in S$.

I surmised, therefore, that the characteristic of a direct sum of $n$ rings must be the $lcm$ of the characteristics.

This is true since each $char(R_{i})$ must divide $char(R)$ by both the definition of the direct sum and the definition of a ring's characteristic.

However, $char(R)$ is defined to be the least such positive integer, so that $char(R_{i}) \vert char(R)\, \implies \, char(R) = lcm(char(R_{i}))$ for $1 \leq i \leq n$.

But, I'm not sure that this reasoning necessarily constitutes a rigorous proof (seems a bit too hand-wavey to me).

Is there a way I could make this proof more rigorous? If so, how would I go about doing so?

Thanks in advance.

Answer 1

The following are defining properties for characteristic and lowest common multiple:

• $mx=0$ for all $x\in R\iff m$ is a multiple of $\mathrm{char}(R)$
• $m$ is a multiple of $c_1,\cdots,c_n\iff m$ is a multiple of $\mathrm{lcm}(c_1,\cdots,c_n)$

By "defining" properties, I mean these properties uniquely characterize the value of a ring's characteristic and the LCM of a set of numbers. In the context of category theory, we would call these "universal properties."

Knowing these facts, to show $\mathrm{char}(R_1\oplus\cdots\oplus R_n)=\mathrm{lcm}(\mathrm{char}(R_1),\cdots,\mathrm{char}(R_n))$, it suffices to observe that the following are equivalent:

• $m(x_1,\cdots,x_n)=(0,\cdots,0)$ for all $(x_1,\cdots,x_n)\in R_1\oplus\cdots\oplus R_n$
• for each $i=1,\cdots,n$ we have ($mx=0$ for all $x\in R_i$)
• for each $i=1,\cdots,n$ we have ($m$ is a multiple of $\mathrm{char}(R_i)$)
• $m$ is a multiple of $\mathrm{lcm}(\mathrm{char}(R_1),\cdots,\mathrm{char}(R_n))$

The fact that the first and last bullet point are equivalent tells us $\mathrm{lcm}(\mathrm{char}(R_1),\cdots,\mathrm{char}(R_n))$ satisfies the universal property for $\mathrm{char}(R_1\oplus\cdots\oplus R_n)$, hence (by uniqueness) must be equal.

(The characteristic of a ring is the exponent of its underlying additive group. The above applies to exponents of products of not-necessarily-commutative groups, even infinitely many.)