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 CE 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.

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 CE 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.

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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