Rng (algebra)
Algebraic ring without a multiplicative identity / From Wikipedia, the free encyclopedia
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In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng (IPA: /rʌŋ/) is meant to suggest that it is a ring without i, that is, without the requirement for an identity element.[1]
There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see Ring (mathematics) § History). The term rng was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity.
A number of algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space.