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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 back to
1828, in which he proved that elements in the ri
John A. Beachy
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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 Artinian ring.
Solution: Let J be the Jacobson radical o
John A. Beachy
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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: Suppose that the cosets x1 , . . . , xk generate M/N
John A. Beachy
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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 clear that 0 i=1 Mi , and so the union is nonempty. Let x,
John A. Beachy
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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 I = ker(), where : R F is a ring homomorphism, and F i
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 (x) = am xm + . . . + a1 x + a0 , g(x) = bn xn + . . .
J.A.Beachy
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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 Ive already given
for modules, to keep this section more
John A. Beachy
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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, then there
exists a1 R with aa1 = 1 and a1 a = 1. Then (a
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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 linear algebra
course. If you took the course at the so
J.A.Beachy
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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 intermediate
level, between high school and university, and after
John A. Beachy
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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 proof has to involve the distributive laws, because they