Zero element

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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.