Rational Mathematics

by G. P. Jelliss

Rings

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.