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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 36E: 36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of...

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22. Suppose that R is a commutative ring with identity. If I and J are ideals of
R for which R/I≈ R/J as R-modules, then prove that I = J. Is the
result true if R/TR/J as rings?

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14. Let be an ideal in a ring with unity . Prove that if then .
Exercises If and are two ideals of the ring , prove that is an ideal of .
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.
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Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.
Prove that if R is a field, then R has no nontrivial ideals.

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