The identity element of an additive monoid or group or of any other algebraic structure (e.g., ring, module, abstract vector space, algebra) equipped with an addition. It is also called the additive identity and is denoted 0. The name and the symbol are borrowed from the ring of integers whose additive identity is, of course, number 0.
The zero element of a ring 
has the property that 
for all 
and, moreover, for every element 
of an 
-module 
it holds that 
. Here, the indices
distinguish the zero element of the ring from the zero element of the module. The
latter also fulfils the rule 
for all 
.
The notation 0 is sometimes also used for the universal bound 
of a Boolean algebra 
. In fact it behaves with respect to the operation 
like a zero element with respect to
multiplication, since 
for all 
.
