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.
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.
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) < ∞).
Another example is the rng of all real sequences that converge to 0, with component-wise operations.
In particular, the real-valued continuous functions with compact support defined on some topological space, together with pointwise addition and multiplication, form a rng; this is not a ring unless the underlying space is compact.
In 2Z, the only multiplicative idempotent is 0, the only nilpotent is 0, and the only element with a reflexive inverse is 0.
equipped with coordinate-wise addition and multiplication is a rng with the following properties:
Since j is injective, we see that R is embedded as a (two-sided) ideal in R^ with the quotient ring R^/R isomorphic to Z.
[3] The process of adjoining an identity element to a rng can be formulated in the language of category theory.
The construction of R^ given above yields a left adjoint to the inclusion functor I : Ring → Rng.
Notice that Ring is not a reflective subcategory of Rng because the inclusion functor is not full.
There are several properties that have been considered in the literature that are weaker than having an identity element, but not so general.
If the associative algebra A over the field K is not unital, one can adjoin an identity element as follows: take A × K as underlying K-vector space and define multiplication ∗ by for x, y in A and r, s in K. Then ∗ is an associative operation with identity element (0, 1).