The rings Z n form a class of commutative rings that is a good source of examples and counterexamples. Definition Let S be a commutative ring. A nonempty subset R of S is called a subring of S if it is a commutative ring under the addition and multiplication of S. 5.1.3. Definition Let R be a commutative ring with identity element 1. The element a is also called a unit of R, and its multiplicative inverse is usually denoted by a -1 .
Proposition Let R be a commutative ring with identity.
Definition Let R and S be commutative rings. An isomorphism from the commutative ring R onto itself is called an automorphism of R. 5.2.2. The inverse of a ring isomorphism is a ring isomorphism. The composition of two ring isomorphisms is a ring isomorphism. S. Example 5.2.5.