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6.1. A RECOLLECTION OF RINGS
269
Exercises 6.1
6.1.1.
Show that if a ring
R
has a multiplicative identity, then the mul
tiplicative identity is unique. Show that if an element
r
2
R
has a left
multiplicative inverse
r
0
and a right multiplicative inverse
r
00
, then
r
0
D
r
00
.
6.1.2.
Verify that
RŒx
1
;:::;x
n
Ł
is a ring for any commutative ring
R
with
multiplicative identity element.
6.1.3.
Consider the set of inﬁnitebyinﬁnite matrices with real entries that
have only ﬁnitely many nonzero entries. (Such a matrix has entries
a
ij
,
where
i
and
j
are natural numbers. For each such matrix, there is a natural
number
n
such that
a
ij
D
0
if
i
±
n
or
j
±
n
.) Show that the set of such
matrices is a ring without identity element.
6.1.4.
Show that (a) the set of upper triangular matrices and (b) the set of
upper triangular matrices with zero entries on the diagonal are both sub
rings of the ring of all
n
by
n
matrices with real coefﬁcients. The second
example is a ring without multiplicative identity.
6.1.5.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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