Math 250A, Fall 2004
Last Midterm Exam—November 4, 2004
Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs,
and other electronic devices. You may refer to a single 2sided sheet of notes. Explain your answers
in full English sentences as is customary and appropriate. Your paper is your ambassador when it
is graded.
All rings are rings with identity!
1.
Suppose that
I
and
J
are ideals in a commutative ring
R
such that
I
+
J
=
R
.
Establish the
surjectivity of the natural map
R
→
R/I
×
R/J,
r
→
(
r
mod
I, r
mod
J
)
. (Don’t just name a
theorem; write down a complete proof.)
This is the Chinese Remainder Theorem, but we are asked to supply a proof. If
I
+
J
=
R
, then 1
is an element of
I
+
J
, so that there are
x
∈
I
,
y
∈
J
with
x
+
y
= 1. Given
a, b
∈
R
, we can write
down
r
:=
ay
+
bx
and see that
r
has the same image as
a
mod
I
and the same image as
b
mod
J
.
If
I
+
J
=
R
, as above, show that
I
n
+
J
m
=
R
whenever
n
and
m
are positive integers.
We can take
m
=
n
since
I
n
+
J
m
contains
I
n
+
J
n
if
n
≥
m
. Suppose that 1 =
x
+
y
as above.
Then 1 = (
x
+
y
)
2
n
. If you expand out (
x
+
y
)
2
n
by the binomial theorem, you’ll see that each
term is divisible either by
x
n
or by
y
n
. Hence (
x
+
y
)
2
n
lies in
I
n
+
J
n
.
2.
Let
k
be a field, and let
V
be the
k
vector space consisting of
(
a
1
, a
2
, . . .
)
with
a
i
∈
k
and
a
i
non
zero only for a finite set of
i
. Let
R
= End
k
V
be the ring of linear transformations
V
→
V
; the
ring multiplication is composition. (If
V
were instead the smaller vector space
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 tao/analysis
 Math, Ring, Commutative ring, free abelian group, Principal ideal domain

Click to edit the document details