Show that If R is a commutative ring with unity and a e R, then(a)%3D {ar:r ER}=aR.

Answered: Image /qna-images/answer/3699017d-1614-42cb-b43c-ef0bd428f7d6.jpg

Answered: If R is a commutative ring with unity…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 15E: 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .

See similar textbooks

Similar questions

15. Let and be elements of a ring. Prove that the equation has a unique solution.
Find the characteristic of each of the following ring: a. b. c. M2() d. M2() e. M2(2) f. M2(3)
19. Find a specific example of two elements and in a ring such that and .
28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.
Given that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.
37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
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
44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Exercises If and are two ideals of the ring , prove that is an ideal of .
Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R is a commutative ring with unity with respect to matrix addition and multiplication. Answer the following questions and give a reason for any negative answers. Is 12 an integral domain? Is R a field?

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.