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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 35E: Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a...

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Qa: prove that
1. Let R be a commutative ring with identity and I be a maximal ideal of R. Then the
quotient ring R/I is a field.
2. Every group of order less than or equal to 5 is abelain.

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18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
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.)
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 .)
14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.
Exercises If and are two ideals of the ring , prove that is an ideal of .
Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.
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
Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]
27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.

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