ABSTRACT
This chapter considers an algebraic structure that has two binary operations and builds off the idea of a group. It provides the definition of a ring and shows that in a ring, the additive operation must be commutative, but the multiplicative operation need not be commutative. The chapter also provides a number of examples of rings and proves that in any finite ring, every non-zero element is either invertible or a zero divisor. As in the case of groups, some rings have the property that some of their substructures satisfy very nice conditions. A mapping between two rings is a homomorphism if it preserves both the additive and the multiplicative operations. The chapter gives a few examples of ring homomorphisms. An integral domain is a commutative ring with unity, which does not contain any zero divisors. One of the interesting properties of integral domains is a kind of cancellation property.
