nLab
rng

Rngs

Idea

A rng 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.

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.

Created on September 9, 2010 16:56:56 by Toby Bartels (64.89.61.238)