College of the Holy Cross, Fall 2009Math 351 – Homework 2 selected solutionsChapter 13, #28a.SinceRis an abelian group under addition, we know from the FundamentalTheorem of Finite Abelian groups that under additionRis isomorphic toZ6. Note that this doesnot necessarily tell us thatRis isomorphic toZ6as a ring (that is, with respect to multiplication aswell). However, we know thatRis cyclic under addition, so there is an elementx∈Rof additiveorder 6. Thusx+x+x+x+x+x= 0, but no smaller sum ofxwith itself is 0. This meansthat the characteristic ofRis at least 6. On the other hand, every element ofRhas additive orderdividing 6 (by Lagrange’s Theorem and the fact thatRhas only 6 elements), so the characteristicofRis at most 6. ThusRhas characteristic 6. Since this is not prime,Rcannot be an integraldomain by Theorem 13.4.Chapter 13, #42.Suppose the characteristic ofRis a primep, and thata∈Ris nilpotent.
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