## Rings and Ideals

This chapter takes up the study of general rings and their homomorphisms, shows how the latter are associated with ideals. It applies the concept of ideals to the geometry of algebraic curves and surfaces. For every homomorphism of a ring there is a corresponding ideal of elements mapped on zero. Conversely, given an ideal, we shall now construct a corresponding homomorphic image. An ideal C in a ring A is a subgroup of the additive group of A. Each element a in A belongs to a coset, often called the residue class a' = a + C, which consists of all sums a + c for variable c in C. In a noncommutative ring one may consider "one-sided" ideals. Since a field is defined as an integral domain in which division (except by zero) is possible, the discussion of characteristics applies at once to fields.