Ring
A Ring is a set with two binary operators such that
- under the first binary operator, the set is a abelian group
- under the second binary operator, the set is a monoid
- the second binary operator distributes over the first
Some authors use terminology that a “ring with inverse” is separate from a “ring.” Some will say “rng” to denote the absence of an inverse.
Field
A field is a set with two binary operators such that
- under the first binary operator, the set is an abelian group
- under the second binary operator, excluding the zero element, the set is an abelian group
A finite field (over a prime number) is sometimes called a Galois field.