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

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

The characteristic of $R$ is

$\mathrm{char}(R):=\{\begin{array}{ll}\min T_R& \text{if }{T}_R\mathrm{\ne }\mathrm{\varnothing },\\ 0& \text{if }{T}_R=\mathrm{\varnothing }\mathrm{.}\end{array}$

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 $\mathrm{char}(R)$ 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 $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity writes $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity. Here the value in the corresponding case is the natural number $0$, not a symbol $\mathrm{\infty }$. A reader coming straight from The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ 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 $\mathrm{char}(R)\mid n$; and in an integral domain every nonzero element has the same additive order as $1_R$ uniform across the two cases.

$\mathrm{char}(R)$ 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 $\mathrm{char}(R)\mathrm{\notin }R$ 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 $\mathrm{char}(R)=1$, 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 $\mathrm{char}(R)$ 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 $\mathrm{char}(R)\mid n$; 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 $\mathrm{char}(R)\mid n$; 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.