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 .