1. Let R be a commutative ring and a any element in R. Define the annhilator of a to be the set ann(a) = {r € R | ra = 0}(that is, the set of all elements that multiply ato zero). Prove that ann(a)is an ideal of R. 2. Referring to problem 1, we will calculate the annihilators of elements in various rings: (a) Let R = Z6 (the integers modulo 6) - determin ann([2]) and ann([5]) (that is, the annihilators of the classes [2] and 5]) (b) Let R Z18 (the integers modulo 18) - determine ann([6]) (express it as a principal ideal generated by an element in the fing) = (c) Let R = Z× Z - determine ann((1,0)) (that is, the annihiltor of the ordered pair (1,0))

1. Let R be a commutative ring and a any element in R. Define the annhilator of a to be the set ann(a) = {r € R | ra = 0}(that is, the set of all elements that multiply ato zero). Prove that ann(a)is an ideal of R. 2. Referring to problem 1, we will calculate the annihilators of elements in various rings: (a) Let R = Z6 (the integers modulo 6) - determin ann([2]) and ann([5]) (that is, the annihilators of the classes [2] and 5]) (b) Let R Z18 (the integers modulo 18) - determine ann([6]) (express it as a principal ideal generated by an element in the fing) = (c) Let R = Z× Z - determine ann((1,0)) (that is, the annihiltor of the ordered pair (1,0))

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 31E: Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set...

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