ring,
n. 1. the area between two concentric circles, an annulus.
2a. (UK usage) a non-empty set endowed with two binary operations, usually called addition and multiplication, such that the set is an Abelian group under the addition and a semi-group under the multiplication, the latter being both left- and right-distributive over addition. If, furthermore, the ring has a multiplicative identity element, it is said to be a ring with identity; thus, the integers are a ring with identity, but the even integers are not. The possibility of a zero ring is not excluded.
b. (North American usage) as above, with a non-zero identity element. See also commutative ring, division ring, integral domain. Compare group, field.
2a. (UK usage) a non-empty set endowed with two binary operations, usually called addition and multiplication, such that the set is an Abelian group under the addition and a semi-group under the multiplication, the latter being both left- and right-distributive over addition. If, furthermore, the ring has a multiplicative identity element, it is said to be a ring with identity; thus, the integers are a ring with identity, but the even integers are not. The possibility of a zero ring is not excluded.
b. (North American usage) as above, with a non-zero identity element. See also commutative ring, division ring, integral domain. Compare group, field.