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## CHAPTER 10

### Ideals and homomorphisms

The concept of an “ideal” in a ring is analogous to the concept of a “normal subgroup” in a group. Rings modulo an ideal are constructed in the same canonical fashion as groups modulo a normal subgroup. The role of an ideal in a “homomorphism between rings” is similar to the role of a normal subgroup in a “homomorphism between groups.” Theorems proved in this chapter on the direct sum of ideals in a ring and on homomorphisms between rings are parallel to the corresponding theorems for groups proved in Chapters 5 and 8.

#### 1 Ideals

Definition. A nonempty subset S of a ring R is called an ideal of R if

(i) implies

(ii) and imply and .

Definition. A nonempty subset S of a ring R is called a right (left) ...

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