23. If N denotes the set of noninvertible elements of R, prove that the followingconditions are equivalent:a) (N,+,) is an ideal of (R,+,),b) the quotient ring (R/N,+,) is a field,e) (R, +,) is a local ring.

Given N denote the set of non-invertible elements of R.

Answered: IN denotes the set of noninvertible ele…

IN denotes the set of noninvertible ele conditions are equivalent: (N,+,) is an ideal of (R,+, ), p) the quotient ring (R/N,+,) is a field, -) (R,+,) is a local ring.

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 24E: 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set...

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23. If N denotes the set of noninvertible elements of R, prove that the following
conditions are equivalent:
a) (N,+,) is an ideal of (R,+,),
b) the quotient ring (R/N,+,) is a field,
e) (R, +,) is a local ring.

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