nLab
rng

Rngs

Idea

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 RR with operations of addition and multiplication, such that:

  • RR is a semigroup under multiplication;
  • RR is an abelian group under addition;
  • multiplication distributes over addition.

Fancy definition

More sophisticatedly, we can say that, just as a ring is a monoid object in Ab, so a rng is a semigroup object in AbAb.

References

Nonunital ring theory

A survey of commutative rng theory is in

  • 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)