83
(viii)
If
X
is an in
fi
nite set, then the family of all
fi
nite subsets of
X
forms a subring of the Boolean ring
B
(
X
)
.
Solution.
True.
3.2
Prove that a commutative ring
R
has a unique one 1; that is, if
e
∈
R
satis
fi
es
er
=
r
for all
r
∈
R
, then
e
=
1.
Solution.
Assume that
er
=
r
for all
r
∈
R
. In particular,
e
1
=
1. On the
other hand, the de
fi
ning property of 1 gives
e
1
=
e
. Hence, 1
=
e
.
3.3
Let
R
be a commutative ring.
(i)
Prove the additive cancellation law.
Solution.
Absent.
(ii)
Prove that every
a
∈
R
has a unique additive inverse: if
a
+
b
=
0
and
a
+
c
=
0, then
b
=
c
.
Solution.
Absent.
(iii)
If
u
∈
R
is a unit, prove that its inverse is unique: if
ub
=
1 and
uc
=
1, then
b
=
c
.
Solution.
Absent.
3.4
(i)
Prove that subtraction in
Z
is not an associative operation.
Solution.
In
Z
,
(
a
−
b
)
−
c
=
a
−
(
b
−
c
)
=
a
−
b
+
c
as long as
c
=
0.
(ii)
Give an example of a commutative ring
R
in which subtraction is
associative.
Solution.
If
R
=
F
2
, subtraction is the same as addition, and so it
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 KeithCornell
 Addition, Commutative ring

Click to edit the document details