Characteristic (algebra)

In mathematics, the characteristic of a ring

R

{\displaystyle R}

, often denoted

char

⁡

(

R

)

{\displaystyle \operatorname {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

)

{\displaystyle \operatorname {char} (R)}

is the smallest positive number

n

{\displaystyle n}

such that

1

+

⋯

+

1

⏟

n

summands

=

0

{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ summands}}}=0}

if such a number

n

{\displaystyle n}

exists, and

0

{\displaystyle 0}

otherwise.

Motivation

The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.

The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer

n

{\displaystyle n}

such that:

a

+

⋯

+

a

⏟

n

summands

=

0

{\displaystyle \underbrace {a+\cdots +a} _{n{\text{ summands}}}=0}

for every element

a

{\displaystyle a}

of the ring (again, if

n

{\displaystyle n}

exists; otherwise zero). This definition is equivalent for a ring, because of distributivity. For rngs (rings without identity), the former definition is nonsensical, and the latter definition is generally used.

Integers, rational numbers and real numbers have characteristic 0.

The integers modulo n have characteristic

n

{\displaystyle n}

.

Every Boolean ring has characteristic 2.