Serge Ballif
MATH 535 Homework 11
November 30, 2007
12) Let
I
be an ideal of the ring
R
, and let
M, N
be
R
modules such that
IN
= 0
. Then show that, as
R/I
modules,
M
⊗
R
N
’
M/IM
⊗
R/I
N
with the isomorphism “given by”
m
⊗
R
n
7→
[
m
]
⊗
R/I
n
.
Remark: Be sure to base your argument on the Universal Mapping Property of tensor products. For example, you may
induce maps between the two sides via their universal mapping properties and show that the isomorphism satisfies
the property stated. Or you may simply show that one object satisfies the Universal Mapping Property of the other.
(Recall the strict warning against trying to define maps from
M
⊗
R
N
by what they do on elements
m
⊗
R
n
.) Although
nothing much exciting is going on, it is beneficial to work through such an example to gain experience in dealing with
tensor products.
Claim 1.
M
⊗
R
N
is an
R/I
module.
Proof.
Scalar multiplication is this case is given by (
r
+
I
)(
m
⊗
n
) =
rm
⊗
n
=
m
⊗
rn
. This map is well defined because for
r
+
i
=
r
0
we have
[
r
0
](
m
⊗
n
) = (
r
+
i
)(
m
⊗
n
)
= (
rm
+
im
)
⊗
n
=
rm
⊗
n
+
im
⊗
n
=
rm
⊗
n
+
m
⊗
in
=
rm
⊗
n
= [
r
](
m
⊗
n
)
.
M
⊗
R
N
satisfies the properties of being an
R/I
module as a consequence of the
fact that it is an
R
module.
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 Fall '08
 BROWNAWELL,WOODRO
 Math, Algebra, Vector Space, Category theory, Tensor, M R N, M/IM R/I, Z2 Z2

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