# Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity 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)

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

if such a number n exists, and 0 otherwise.