A set R is a **ring** with respect to operations O and I if it is a
commutative group with respect to O, a semigroup with respect to I (i.e. closed and
associative) and I is distributive over O on right and left. We denote the O-identity by o.
Then oIx = o = xIo for all x in R, and –(aIb) = (–a)Ib = aI(–b), where – denotes the O
inversion.

A **subring** is a subset of a ring which is also a ring with respect to the same operations.
An **ideal** of a ring R is a subset S which is an O-group and has the property that if r∈R and s∈S then (rIs)∈S.
The intersection of two ideals is an ideal. The 'sum' of two ideals is an ideal, i.e. if U and V are ideals then so is UOV,
meaning the set of all elements uOv with u in U and v in V.

A **commutative ring** has I commutative. In a commutative ring R, the set of all
elements aIr for fixed a and variable r is an ideal of R.

A **ring with identity** has an I-identity i. In such a ring, (–i)Ix = –x, i.e.
the O-inverse operator is equivalent to I-multiplication by the O-inverse of the I-identity.
It is possible for a ring with identity to contain a subring with identity, but with the two identities different.
If U is an ideal of a ring R with identity i and i is in U then U=R.

An **integral domain** is a commutative ring with identity and no 'divisors of zero',
that is such that xIy = o implies x=o or y=o or both. Equivalently an integral
domain is a commutative ring with identity in which **cancellation** holds, that
is aIb = aIc implies b=c if a is not o.

A **field** is an integral domain in which every nonzero element has a multiplicative
inverse in the field. The only ideals of a field are {o} and the field itself. A finite
integral domain is a field.