A rng (terminology due to Jacobson) is a ring ‘without identity’ (hence the missing ‘i’ in the name, get it?). By the red herring principle, we sometimes speak of a nonunital ring. Note that classically, the word ‘ring’ originally meant a rng, but we usually require our rings to have identities.

Definitions

Explicit definition

Specifically, a rng is a set$R$ with operations of addition and multiplication, such that:

D. Anderson, Commutative rngs, in J. Brewer et al. (eds.) Multiplicative ideal theory in Commutative Algebra, 2006

Terminology

The notation “rng” originates in

Nathan Jacobson Basic Algebra,

where the term is attributed to Louis Rowen.

(Bourbaki 6, chapter 1) uses the term “pseudo-ring” instead, which however has not caught on and even if more sane, will be understood less than “rng”.

Revised on August 19, 2014 17:20:15
by Toby Bartels
(75.88.46.170)