The ring Z n is isomorphic to the direct sum of the rings Z k that arise in the prime factorization of n. This structure theorem can be used to determine the invertible, idempotent, and nilpotent elements of Z n and provides an easy proof of our earlier formula for the Euler phi-function in terms of the prime factors of n. 5.2.9. Definition Let R be a commutative ring with identity. If no such positive integer exists, then R is said to have characteristic zero . Ideals and factor rings 5.3.1.
Definition Let R be a commutative ring.
Proposition Let R be a commutative ring with identity. Then R is a field if and only if it has no proper nontrivial ideals. Integral domains 5.1.6. The ring of integers Z is the most fundamental example of an integral domain. The ring of all polynomials with real coefficients is also an integral domain, but the larger ring of all real valued functions is not an integral domain.