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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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