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Let N be a group that is closed under the operation of addition, denoted +. An additive identity for N, denoted e, is an element in N such that for any element n in N,
e + n = n = n + e.
In a group, the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof).
A ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below).
In the ring Mm × n(R) of m by nmatrices over a ring R, the additive identity is the zero matrix, denoted O or 0, and is the m by n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 × 2 matrices over the integers M2(Z) the additive identity is