Rings, Integral Domain

Integral Domain

A domain is a ring with left and right cancellation. That is, a*x = a*y implies x = y when a is nonzero.

An equivalent definition says there are no zero divisors. If cancellation fails we have b*(x-y) = 0, making b and x-y zero divisors. (This is where semirings start to veer off, since there is no such thing as x-y.) Conversely, if we have zero divisors, b*x = b*0, and cancellation fails.

An integral domain is a commutative domain. It looks somewhat like the integers.

If x and y are associates in an integral domain, write xu = y and yv = x. Thus xuv = x, and by cancellation, uv = 1, making u and v units. In an integral domain, each set of associates contains one element for every unit. Each set has the same cardinality, namely the cardinality of the units. Zero is of course an exception to the rule. Its associates are all zero.

The integers have units ±1, and every nonunit has one other associate, its opposite.

Finite Domain is a Field

Every finite domain is a division ring, and consequently, a field.

Given a nonzero element c, c*x is a permutation on the ring by cancellation. A 1-1 map from a finite set into itself is onto, so c maps some element d to 1. This means cd = 1, hence c is right invertible. Since c was arbitrary, all elements are right invertible, and by an earlier result, the ring is a division ring.

Every finite division ring is a field. combine these results and every finite domain is a field.