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.

Revised on June 19, 2014 08:32:35 by Colin Zwanziger (74.92.19.28)