In algebra, the term zero element is used with two meanings, both in analogy to the number zero.
- For an additively written binary operation, z is a zero element if (for all g)
- z + g = g = g + z
- i.e., it is the (unique) neutral element for this operation.
- For a multiplicatively written binary operation, a is a zero element if (for all g)
- ag = g = ga
- i.e., it is the (unique) absorbing element for this operation.
In rings (not only the real or complex numbers) 0 is the zero element in both senses.
In addition to these "two-sided" zero elements, (one-sided) left or right zero elements are also considered for which only one of the two identities is valid for all g.
One-sided zero elements need not be unique.