Characteristic (algebra)

In mathematics, the characteristic of a ring $R$ , often denoted $char(R)$ , is defined to be the smallest positive number of copies of the ring's multiplicative identity ( $1$ ) that will sum to the additive identity ( $0$ ). If no such number exists, the ring is said to have characteristic zero.

That is, $char(R)$ is the smallest positive number $n$ such that:^[1]^{(p 198, Thm. 23.14)}

if such a number $n$ exists, and $0$ otherwise.

The characteristic is the $n$ such that $\mathbb {Z}$ is the kernel of the unique ring homomorphism from $\mathbb {Z}$ to $R$ .^[a]

natural number

The characteristic is the $n$ such that $R$ contains a subring isomorphic to the factor ring $\mathbb {Z} /n\mathbb {Z}$ , which is the image of the above homomorphism.

natural number

When the non-negative integers ${0, 1, 2, 3, ...}$ are by divisibility, then $1$ is the smallest and $0$ is the largest. Then the characteristic of a ring is the smallest value of $n$ for which $n \cdot 1 = 0$ . If nothing "smaller" (in this ordering) than $0$ will suffice, then the characteristic is $0$ . This is the appropriate partial ordering because of such facts as that $char(A \times B)$ is the least common multiple of $char A$ and $char B$ , and that no ring homomorphism $f : A \to B$ exists unless $char B$ divides $char A$ .

partially ordered

The characteristic of a ring $R$ is $n$ precisely if the statement $ka = 0$ for all $a \in R$ implies that $k$ is a multiple of $n$ .

Case of rings[edit]

If $R$ and $S$ are rings and there exists a ring homomorphism $R \to S$ , then the characteristic of $S$ divides the characteristic of $R$ . This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic $1$ is the zero ring, which has only a single element $0$ . If a nontrivial ring $R$ does not have any nontrivial zero divisors, then its characteristic is either $0$ or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic $0$ is infinite.

The ring $\mathbb {Z} /n\mathbb {Z}$ of integers modulo $n$ has characteristic $n$ . If $R$ is a subring of $S$ , then $R$ and $S$ have the same characteristic. For example, if $p$ is prime and $q (X)$ is an irreducible polynomial with coefficients in the field $\mathbb {F} _{p}$ with $p$ elements, then the quotient ring $\mathbb {F} _{p}[X]/(q(X))$ is a field of characteristic $p$ . Another example: The field $\mathbb {C}$ of complex numbers contains $\mathbb {Z}$ , so the characteristic of $\mathbb {C}$ is $0$ .

A $\mathbb {Z} /n\mathbb {Z}$ -algebra is equivalently a ring whose characteristic divides $n$ . This is because for every ring $R$ there is a ring homomorphism $\mathbb {Z} \to R$ , and this map factors through $\mathbb {Z} /n\mathbb {Z}$ if and only if the characteristic of $R$ divides $n$ . In this case for any $r$ in the ring, then adding $r$ to itself $n$ times gives $nr = 0$ .

If a commutative ring $R$ has prime characteristic $p$ , then we have $(x + y) p = x p + y p$ for all elements $x$ and $y$ in $R$ – the normally incorrect "freshman's dream" holds for power $p$ . The map $x \mapsto x p$ then defines a ring homomorphism $R \to R$ , which is called the Frobenius homomorphism. If $R$ is an integral domain it is injective.