multiplicative identity
Noun
Meaning
multiplicative identity (plural multiplicative identities)
(algebra) An element of an algebraic structure, generally denoted 1, which is an identity for a multiplicative operation (generally denoted × or *, or by concatenation).
Examples
An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1).
Also, the multiplicative inverse is the reciprocal of any number (except 0; 0 is the only number without a multiplicative inverse), that is, multiplying the reciprocal of any number by the number itself yields the multiplicative identity.
A ring object in C is an object R equipped with morphisms (addition), (multiplication), (additive identity), (additive inverse), and (multiplicative identity) satisfying the usual ring axioms.
A structure satisfying all the axioms except possibly the existence of a multiplicative identity 1 is called a rng (or sometimes pseudo-ring ).
In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.
Non-unital algebras Some authors use the term "associative algebra" to refer to structures with do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.