The characteristic of a ring: the least $n \ge 1$ with $n \cdot 1_R = 0$ when one exists, and $0$ otherwise

Let $R$ be a ring (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides) with identity $1_R$ and zero $0_R$. For $n \in \mathbb{N}$ let $n \cdot 1_R$ be the additive natural power of $1_R$ in the abelian group $(R,+,0_R)$ (Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$, Group and abelian group): thus $0 \cdot 1_R = 0_R$ and $\sigma(n)\cdot 1_R = n \cdot 1_R + 1_R$. Put

$T_R \;:=\; \{\, n \in \mathbb{N} \;:\; n \ge 1 \text{ and } n \cdot 1_R = 0_R \,\} \;\subseteq\; \mathbb{N}.$

The characteristic of $R$ is

$\operatorname{char}(R) \;:=\; \begin{cases} \min T_R & \text{if } T_R \ne \varnothing, \\[2pt] 0 & \text{if } T_R = \varnothing.\end{cases}$

Why the least element exists. $T_R$ is a subset of $\mathbb{N}$, so when it is nonempty it has a least element by the well-ordering principle (The well-ordering principle), and that element is unique: two least members are below one another and hence equal by antisymmetry of the natural-number order ($\le$ is a linear order on $\mathbb{N}$, Order on the natural numbers). So $\operatorname{char}(R)$ is a well-defined natural number in both cases.

The clause $n \ge 1$ is not decoration. $\mathbb{N}$ contains $0$ (The natural numbers $\mathbb{N}$ (von Neumann)) and $0 \cdot 1_R = 0_R$ holds in every ring, so without that clause $T_R$ would always contain $0$ and the definition would say nothing.

Convention: the value in the empty case is $0$, and this is the OPPOSITE of the convention for the order of a group element. The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity writes $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity. Here the value in the corresponding case is the natural number $0$, not a symbol $\infty$. A reader coming straight from The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity should notice the difference: it is deliberate, it is the standard convention for the characteristic, and it is what makes the divisibility statement of The characteristic of a ring is the additive order of $1_R$, with $0$ recording infinite order; $n \cdot 1_R = 0$ holds exactly when $\operatorname{char}(R) \mid n$; and in an integral domain every nonzero element has the same additive order as $1_R$ uniform across the two cases.

$\operatorname{char}(R)$ is a natural number, hence a set, not an element of $R$. A natural number in this library is a von Neumann natural (The natural numbers $\mathbb{N}$ (von Neumann)), so $\operatorname{char}(R) \notin R$ in general and the expression $n \cdot 1_R$ is not a product in $R$ but the additive multiple just described.

Dictionary for fields. When $R$ is a field $F$ (Field), the element $n \cdot 1_F$ is exactly the canonical natural $\iota(n)$ of The canonical natural $\iota(n) = n \cdot 1_F$ of a field; this is proved, not assumed, in In a field, the additive multiple $n \cdot 1_F$ is the canonical natural $\iota(n)$: the additive power of the group-power definition and the canonical natural are the same function, both being the unique one given by the recursion $\iota(0) = 0_F$, $\iota(\sigma(n)) = \iota(n) + 1_F$. So for a field $T_F = \{\, n \ge 1 : \iota(n) = 0_F \,\}$ and the characteristic is the least $n \ge 1$ with $\iota(n) = 0_F$, or $0$ if there is none.

• Characteristic $1$ occurs, exactly once. $1 \cdot 1_R = 0 \cdot 1_R + 1_R = 1_R$, so $1 \in T_R$ exactly when $1_R = 0_R$, that is exactly when $R$ is the one-element ring; and then $\operatorname{char}(R) = 1$, since $1$ is the least member of $\mathbb{N}$ with $n \ge 1$. The companion page records that ring. Every other ring has characteristic $0$ or a value at least $2$.

• Characteristic $0$ means "no such $n$", not "$n = 0$ works". The value $0$ is a flag, and the flag is chosen so that "$n \cdot 1_R = 0_R$ exactly when $\operatorname{char}(R)$ divides $n$" is true in both cases: when the characteristic is $0$ the right-hand side says $n = 0$, which is exactly right. That statement is The characteristic of a ring is the additive order of $1_R$, with $0$ recording infinite order; $n \cdot 1_R = 0$ holds exactly when $\operatorname{char}(R) \mid n$; and in an integral domain every nonzero element has the same additive order as $1_R$.

• A further theorem about the possible characteristics of an integral domain is not stated here. The current corpus does not yet state or prove that result. A future algebra page may add it from The characteristic of a ring is the additive order of $1_R$, with $0$ recording infinite order; $n \cdot 1_R = 0$ holds exactly when $\operatorname{char}(R) \mid n$; and in an integral domain every nonzero element has the same additive order as $1_R$ once the needed prime-number interface is brought into scope.