Rng (algebra)

In mathematics, and more specifically in abstract algebra, a rng (or pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. 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 history section of the article on rings). 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

Rng homomorphisms are defined in the same way as ring homomorphisms except that the requirement f(1) = 1 is dropped. That is, a rng homomorphism is a function f: R → S from one rng to another such that

for all x and y in R.

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.

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