# 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 $R$ with operations of addition and multiplication, such that

• $R$ is a semigroup under multiplication;
• $R$ is an abelian group under addition;
More sophisticatedly, we can say that, just as a ring is a monoid object in Ab, so a rng is a semigroup object in $\mathrm{Ab}$.