College of the Holy Cross, Fall 2009 Math 351 – Homework 2 selected solutions Chapter 13, #28a. Since R is an abelian group under addition, we know from the Fundamental Theorem of Finite Abelian groups that under addition R is isomorphic to Z 6 . Note that this does not necessarily tell us that R is isomorphic to Z 6 as a ring (that is, with respect to multiplication as well). However, we know that R is cyclic under addition, so there is an element x ∈ R of additive order 6. Thus x + x + x + x + x + x = 0, but no smaller sum of x with itself is 0. This means that the characteristic of R is at least 6. On the other hand, every element of R has additive order dividing 6 (by Lagrange’s Theorem and the fact that R has only 6 elements), so the characteristic of R is at most 6. Thus R has characteristic 6. Since this is not prime, R cannot be an integral domain by Theorem 13.4. Chapter 13, #42. Suppose the characteristic of R is a prime p , and that a ∈ R is nilpotent. Thus there is some
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