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Published in:
2019 | OriginalPaper | Chapter
4. Quotient Structures
Author : Matej Brešar
Published in: Undergraduate Algebra
Publisher: Springer International Publishing
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Abstract
In this chapter, we will get acquainted with special types of subgroups called normal subgroups. With the help of a normal subgroup we can construct a new group called a quotient group. In a similar fashion, we construct quotient rings from certain subsets of rings called ideals. We have already encountered a particular case of these constructions in Section 2.2 on integers modulo n, but at that time we avoided the terms “quotient group” and “quotient ring”. Quotient structures are intimately connected with homomorphisms. Every homomorphism gives rise to a quotient structure, and every quotient structure gives rise to a homomorphism. Through this connection, we will better understand the meaning of homomorphisms that are not injective. However, the real meaning and importance of both homomorphisms and quotient structures will become evident in Part II, where we will use them as tools for solving various problems.