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Concept
Rng (algebra)
Summary
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.
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