College Algebra Exam Review 336

College Algebra Exam Review 336 - 346 8. MODULES that '.r/...

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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 End.M /. 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 following lemma. Lemma 8.1.4. Let M be a module over the ring R. then for all r 2 K and v 2V, (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. I Example 8.1.5. A unital module over a field K is the same as a K –vector space. ...
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