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Unformatted text preview: 346 8. MODULES that '.r/ 2 End.M / and the associative law and ﬁrst distributive law say
that r 7! '.r/ is a ring homomorophism from R to End.M /.
Deﬁnition 8.1.2. A module M over a ring R is an abelian group M together with a product R M ! M satisfying
.r1 r2 /m D r1 .r2 m/;
.r1 C r2 /m D r1 m C r2 m; and r.m1 C m2 / D rm1 C rm2 : Deﬁnition 8.1.3. If the ring R has identity element 1, an R–module M is
called unital in case 1m D m for all m 2 M .
The discussion above shows that specifying an R–module M is the
same as specifying a homomorphism ' from R into the endomorphism
ring of the abelian group M . In case R has identity element 1, the R–
module M is unital if, and only if, '.1/ D idM , the identity of the ring
Convention: When R has an identity element, we will assume,
unless otherwise speciﬁed, that all R modules are unital.
We record some elementary consequences of the module axioms in the
Lemma 8.1.4. Let M be a module over the ring R. then for all r 2 K and
(a) 0v D r0 D 0.
(b) r. v / D .rv/ D . r /v .
(c) If R has a multiplicative identity and M is unital, then . 1/v D
v. Proof. This is proved in exactly the same way as the analogous result for
vector spaces, Lemma 3.3.3.
Example 8.1.5. A unital module over a ﬁeld K is the same as a K –vector
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- Fall '08