Page 185 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 5. The only ................. ring homomorphism from Z into itself is Z .
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(a) trivial (b) non-trivial

(c) direct (d) indirect
17.4 Summary

1. The definition of a ring homomorphism, its kernel and its image, along with several
examples.

2. The direct or inverse image of a subring under a homomorphism is a subring.
3. Iff : R - S is a ring homomorphism, then
(i) Im f is a subring of S,
(ii) Ker f is an ideal of R,
(iii) f (1) is an ideal of R for every ideal I of S.
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(iv) iff is surjective, then f(I) is an ideal of S.

4. A homomorphism is injective iff its kernel is {0}.
5. The composition of homomorphisms is a homomorphism.
6. The definition and examples of a ring isomorphism.
7. The proof and applications of the Fundamental Theorem of Homomorphism which says
that iff : R  S is a ring homomorphism, then R/Ker f  Im f.

17.5 Keyword

Isomorphism: If a homomorphism is a bijection, it is called an isomorphism.

17.6 Review Questions

1. Which of the following rings are not fields?

2Z, Z , Z , Q × Q
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2. Will a subring of a field be a field? Why?
3. Show that char (X) = 2, where X is a non-empty set.
4. Let R be a ring and char R = m. What is char (R × R)?

Answers: Self Assessment

1. (c) 2. (a) 3. (b) 4. (b) 5. (b)

17.7 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.

John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

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