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MTHSC 851 (Abstract Algebra)
Dr. Matthew Macauley
HW 9
Due Tuesday April 14th, 2009
(1) Give an example of a ring with exactly 851 ideals.
(2) If
F
is a ﬁeld, show that
M
n
(
F
) is a simple ring.
(3) Let
R
be a ring with unity and
x
∈
R
any nonunit. Use Zorn’s lemma to prove that
x
is
contained in a maximal ideal.
(4) A
local ring
is a commutative ring with identity which has a unique maximal ideal. Prove
that
R
is local if and only if the nonunits of
R
form an ideal.
(5) Let
R
be a ﬁnite ring.
(a) Prove that there are positive integers
m
and
n
with
m > n
such that
x
m
=
x
n
for
every
x
∈
R
. (
Hint
: If

R

=
n
, then consider the ring
S
=
R
×···
R
, with
n
factors.)
(b) Give a direct proof (i.e., without appealing to part (c)) that if
R
is an integral domain,
then it is a ﬁeld.
(c) Suppose that
R
has identity. Prove that if
x
∈
R
is not a zero divisor, then it is a
unit.
(6) (a) An element
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This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.
 Fall '08
 Staff
 Algebra

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