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
+
⋯
+
1
⏟
n
summands
=
0
{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ summands}}}=0}
if such a number n exists, and 0 otherwise.
The special definition of the characteristic zero is motivated by the equivalent definitions given in § Other equivalent characterizations, 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 n such that
a
+
⋯
+
a
⏟
n
summands
=
0
{\displaystyle \underbrace {a+\cdots +a} _{n{\text{ summands}}}=0}
for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity: mandatory vs. optional), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.