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Lecture 1 (Aug 23, 2010)
Throughout this lecture we assume (unless stated otherwise) that
A
is a ring
(not necessarily comutative) with an identity element
1
0
.
Defnition 1.
A left
A
module over
A
is a set
E
with an algebraic structure
given as follows:
1. a commutative group law on
E
(written additively);
2. a scalar multiplication map
A
×
E
→
E,
(
a, x
)
a
·
x
satisfying the fol
lowing axioms:
a
)
a
·
(
x
+
y
) =
a
·
x
+
a
·
y
, for all
a
∈
A, x, y
∈
E
;
b
)
(
a
+
b
)
·
x
=
a
·
x
+
b
·
x
, for all
a, b
∈
A, x
∈
E
;
c
)
a
·
(
b
·
x
) = (
ab
)
·
x
, for all
a,b
∈
A, x
∈
E
;
d
)
1
·
x
=
x
, for all
x
∈
E
.
Replacing c) by
c
′
)
a
·
(
b
·
x
) = (
ba
)
·
x
for all
a, b
∈
A, x
∈
E
, we obtain the deFni
tion of a right
A
module.
Examples:
1. Every commutative group
G
is a
Z
module.
2. If
φ
:
A
→
B
is a ring homomorphism, then we obtain an
A
module
structure on
B
, with the scalar multiplication given by
a
·
b
=
φ
(
a
)
b.
In the special case when
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 Spring '08
 Wilczynski

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