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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...

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4

The intersection of the collection of al
the subrings
containing a given subset M of a ring
R is the smallest
subring containing M.
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42. Let the operation of addition be defined on subsets by. Use a Venn diagram with labelled regions to illustrate each of the following statements. a. b. c. .
Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
In Exercises 15 and 16, the given table defines an operation of multiplication on the set S={ e,a,b,c }. In each case, find a condition in Definition 3.1 that fails to hold, and thereby show that S is not a group. See Figure 3.7 e a b c e e a b c a e a b c b e a b c c e a b c
Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s, (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. Prove that RS is commutative if both R and S are commutative. Prove RS has a unity element if both R and S have unity elements. Given as example of rings R and S such that RS does not have a unity element.
Label each of the following statements as either true or false. Every endomorphism is an epimorphism.
Prove that if m0 and (a,b) exists, then (ma,mb)=m(a,b).

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