Define a divisibility relation | on Zm by this rule: for elements A and B of Zm, A |B if and only if AC=B for some C E Zm- (i) Prove that this relation is transitive. At one point in the proof you will need to use one of the ring or field axioms for Zm. You need not prove that axiom, but write down which axiom it is and state clearly where you are using it. (ii) For which elements of Zm is it true that [2]m | A? Briefly justify your answer. [Hint: do odd m and even m as separate cases.]
Define a divisibility relation | on Zm by this rule: for elements A and B of Zm, A |B if and only if AC=B for some C E Zm- (i) Prove that this relation is transitive. At one point in the proof you will need to use one of the ring or field axioms for Zm. You need not prove that axiom, but write down which axiom it is and state clearly where you are using it. (ii) For which elements of Zm is it true that [2]m | A? Briefly justify your answer. [Hint: do odd m and even m as separate cases.]
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 30E: a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R....
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