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Some history
The study of commutative rings had its origin in three areas: number theory, algebraic
geometry, and invariant theory. The ring Z[i] was used by Gauss in a paper dating b
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CHAPTER 3: STRUCTURE OF NONCOMMUTATIVE RINGS
Review Problems
1. Let R be a left Artinian ring in which I 2 = (0) implies I = (0), for all ideals I of R.
Prove that R is a semisimple A
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CHAPTER 2: MODULES
Review Problems
1. Let M be a left R-module. Show that M is nitely generated if there exists a submodule N M such that N and M/N are both nitely generated.
Solution
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SOLVED PROBLEMS: SECTION 2.1
13. Let M be a left R-module, and let M1 M2 . . . M be an ascending chain of
submodules of M . Prove that i=1 Mi is a submodule of M .
Solution: It is cle
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CHAPTER 1: RINGS
Review Problems
1. Let I be an ideal of the commutative ring R. Prove that I is a prime ideal i I is the
kernel of a ring homomorphism from R into a eld.
Solution: If
Introductory Lectures on Rings and Modules
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SOLVED PROBLEMS: SECTION 1.4
9. Let D be an integral domain. Prove that if the polynomial ring D[x] is a principal
ideal domain, then D is a eld.
10. Let f
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Abelian groups
The goal of this section is to look at several properties of abelian groups and see how they
relate to general properties of modules. Ill usually repeat the denitions I
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SOLVED PROBLEMS: SECTION 1.2
13. Check that any ring homomorphism preserves units, idempotent, and nilpotent elements.
Solution: Let : R S be a ring homomorphism. If a R is a unit, th
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Vector spaces
Im going to begin this section at a rather basic level, giving the denitions of a eld and of
a vector space in much that same detail as you would have met them in a rst
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Introduction
Richard Dedekind was born in 1831 in Braunschweig, in what is now Germany. At age 16
he entered the Collegium Carolinum, where his father taught. It was at an intermediat
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SOLVED PROBLEMS: SECTION 1.1
17. Show that in any ring R the commutative law for addition is redundant, in the sense
that it follows from the other axioms for a ring.
Solution: The pr