The ring characteristic of a unit ring 
, denoted 
, is the least positive
integer 
such that 
, or 0 if no such 
exists. Here 
means the sum of 
copies of the multiplicative
identity. Equivalently, the ring kernel of the
ring homomorphism 
sending 
to 
is 
(Dummit and Foote 2004).
For example, 
has ring characteristic 0, while 
has ring characteristic 
for every positive integer 
. The trivial ring therefore
has ring characteristic 1. For a field, this agrees with
its field characteristic and is either 0
or a prime number. More generally, an integral
domain also has ring characteristic 0 or a prime
number. Indeed, a composite ring characteristic
would give two nonzero elements 
and 
whose product is 0. A local
ring has mixed characteristic 
when its ring characteristic is 0 and its residue
field has field characteristic 
.
