# HW2B - Math103B Homework Solutions HW2 Jacek Nowacki April...

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Math103B Homework Solutions HW2 Jacek Nowacki April 29, 2007 Chapter 14 Problem 6. Find all maximal ideals in a. Z 8 , b. Z 10 , c. Z 12 , d. Z n . Proof. We begin by noticing that each ring in this problem considered as a group is cyclic (this has to do with Z being generated by 1). Now, each ideal, being a subgroup, must also be cyclic (last quarter we proved that all subgroups of a cyclic group are cyclic). a. here each element x is either even in which case x < 2 > , or it is odd x = 2 k + 1 and must be relatively prime to 8. so, is a unit. this way we see that the only maximal ideal is < 2 > . b. here each element x is divisible by either 5, in which case x < 5 > , or it is divisible by 2, in which case x < 2 > , or it is relatively prime to both 5 and 2. being relatively prime to all divisors of 10, it must be relatively prime to 10. hence, a unit. this way we get that < 5 > and < 2 > are the only maximal ideals. c.

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## This note was uploaded on 04/30/2011 for the course MECHANICAL MSC taught by Professor Dr.rahula during the Spring '10 term at University of Moratuwa.

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HW2B - Math103B Homework Solutions HW2 Jacek Nowacki April...

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