Exercise 12.3. Recall that an element a of a ring R is nilpotent if an = 0 for some nЄ N. SupposeR is commutative. Show thatN := {a Є R | a is nilpotent}is an ideal of R, called the nilpotent radical of R.

Answered: Image /qna-images/answer/3244a7d7-064d-425a-9486-5a120b3a5b24.jpg

Answered: Exercise 12.3. Recall that an element a…

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 4E: Exercises If and are two ideals of the ring , prove that is an ideal of .

See similar textbooks

Similar questions

14. Let be an ideal in a ring with unity . Prove that if then .
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 .)
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
Exercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Show that the ideal is a maximal ideal of .
Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
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.
Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .
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
Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.

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

Exercise 12.3. Recall that an element a of a ring R is nilpotent if an = 0 for some nЄ N. Suppose R is commutative. Show that N := {a Є R | a is nilpotent} is an ideal of R, called the nilpotent radical of R.