Rng (algebra)

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, pronounced like rung (IPA: /rʌŋ/), is meant to suggest that it is a ring without i, that is, without the requirement for an identity element.

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.

Rngs appear in the following chain of class inclusions:

Formally, a rng is a set R with two binary operations (+, ·) called addition and multiplication such that

A rng homomorphism is a function f: R → S from one rng to another such that

for all x and y in R.

If R and S are rings, then a ring homomorphism R → S is the same as a rng homomorphism R → S that maps 1 to 1.