S. Example 5.2.5. Proposition Let R and S be commutative rings. Definition Let R and S be commutative rings. 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.