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 (IPA: rʊŋ) 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 ). 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. Formally, a rng is a set R with two binary operations (+, ·) called addition and multiplication such that (R, +) is an abelian group, (R, ·) is a semigroup, Multiplication distributes over addition. A rng homomorphism is a function f: R → S from one rng to another such that f(x + y) = f(x) + f(y) f(x · y) = f(x) · f(y) 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. All rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng. Rngs often appear naturally in functional analysis when linear operators on infinite-dimensional vector spaces are considered. Take for instance any infinite-dimensional vector space V and consider the set of all linear operators f : V → V with finite rank (i.e. dim f(V) < ∞). Together with addition and composition of operators, this is a rng, but not a ring.