HW2_selected_solns - College of the Holy Cross Fall 2009...

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.