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.

**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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