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 
.
