Chapter I . A-module s
Throughout thi s chapte r A wil l denot e a unitar y rin g wit h unit y elemen t 1. A
lomomorphism co: A V o f unitar y ring s wil l alway s b e require d t o ma p unit y elemen t
o unit y element .
Examples, (i ) A = Z, th e ring of integers;
(ii) I f 7 7 i s a group , th e group ring Zn i s th e fre e abelia n grou p wit h n a s basis ,
he multiplicatio n bein g give n o n th e basi s element s b y th e grou p operatio n i n n;
(iii) I f A i s a n abelia n group , the n En d A i s th e ring of endomorphisms of A;
(iv) Give n a unitar y rin g A , w e ma y for m th e opposite ring A o p , t o b e th e rin g
yith th e sam e underlyin g (additive ) abelia n grou p a s A , an d wit h multiplicatio n i n
Vop bein g give n b y
A * p = pX, A , p A .
Definition 1.1 . A left A-module A i s a n abelia n grou p A togethe r wit h a homo -
norphism
co: A— * En d A.
V right A-module i s a lef t A op -module.
We will writ e "A-module " whe n a lef t A-modul e i s intended . I f w e writ e ka fo r
L)(A)(a), the n th e functio n (A , a) +—+ \a satisfie s
(i) A( a + b) = Xa + \b, A 6 A , a , b A,
(ii) ( A + p)a = Xa + pa, A , p 6 A , a £ A,
(iii) {Xp)a = \{pa), A , p 6 A , a €. A,
(iv) l a = a , a €. A.
X is precisel y i n statemen t (iii ) abov e tha t th e distinctio n betwee n lef t an d righ t A -
nodules appears . O f cours e w e writ e a\ i n th e cas e o f a righ t A-module , s o tha t
iii) i s replaced , fo r a righ t A-module , b y
(iii)' a(Xp) = {a\)p, A , p A , a A.
i A i s commutativ e w e mak e n o distinctio n betwee n lef t an d righ t A-modules .
A homomorphism cf: A B o f A-module s i s a homomorphis m o f abelia n group s
satisfying
1
http://dx.doi.org/10.1090/cbms/008/01
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