Let R be a commutative ring and let A be an ideal of R. Show that VA = {xe R: x" e A for some positive integer n} is an ideal of R such that (i) AS VĀ (iii) If R has unity and VA = R, then A = R. (ii) VWA = VA %3D %3D %3D

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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Let R be a commutative ring and let A be an ideal
of R. Show that
VA = {x € R:x" e A for some positive integer n}
is an ideal of R such that
(i) AS VĀ
(iii) If R has unity and VA = R, then A = R.
(ii) VWA = VA
%3D
Transcribed Image Text:Let R be a commutative ring and let A be an ideal of R. Show that VA = {x € R:x" e A for some positive integer n} is an ideal of R such that (i) AS VĀ (iii) If R has unity and VA = R, then A = R. (ii) VWA = VA %3D
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