Modules
Although this chapter is logically self-contained and prepares for future topics, in practice readers will have had some acquaintance with vector spaces over a field. We generalize this notion here to modules over rings. It is a standard fact (to be repro
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III
Modules
Although this chapter is logicall y self-contained and prepares for future topics, in practice readers will have had some acquaintance with vector spaces over a field. We generalize this notion here to modules over rings . It is a standard fact (to be reproved) that a vector space has a basis, but for module s this is not alway s the case . Sometimes they do ; most often they do not. We shall look into cases where they do . For examples of modules and their relations to those which have a basis, the reader should look at the comments made at the end of §4.
§1.
BASIC DEFINITIONS
Let A be a ring . A left module over A, or a left A-module M is an abelian group, usually written additively, together with an operation of A on M (viewing A as a multiplicative monoid by RI 2), such that, for all a, b E A and x, y E M we have (a
+ b)x = ax + bx
and
a(x
+ y) = ax + ay.
We leave it as an exercise to prove that a( - x) = - (ax) and that Ox = O. . By definition of an operation, we have lx = x . In a similar way, one defines a right A-module. We shall deal only with left A-modules, unless otherwise specified, and hence call these simply A-modules, or even modules if the reference is clear.
117 S. Lang, Algebra © Springer Science+Business Media LLC 2002
118
MODULES
III, §1
Let M be an A-module. By a submodule N of M we mean an additive subgroup such that AN c N . Then N is a module (with the operation induced by that of A on M).
Examples We note that A is a module over itself. Any commutative group is a Z-module. An additive group consisting of 0 alone is a module over any ring. Any left ideal of A is a module over A. Let J be a two-sided ideal of A. Then the factor ring AI J is actually a module over A . If a E A and x + J is a coset of J in A , then one defines the operation to be a(x + J) = ax + J. The reader can verify at once that this define s a module structure on AIJ. More general, if M is a module and N a submodule, we shall define the factor module below. Thus if L is a left ideal of A , then AlLis also a module. For more examples in this vein , see §4. A module over a field is called a vector space . Even starting with vector spaces, one is led to con sider modules over rings . Indeed, let V be a vector space over the field K . The reader no doubt already knows about linear maps (which will be recalled below systematically). Let R be the ring of all linear maps of V into itself. Then V is a module over R. Similarly, if V = K" denotes the vector space of (vertical) n-tuples of elements of K , and R is the ring of n x n matrices with components in K , then V is a module over R. For more comments along these line s, see the examples at the end of §2. Let S be a non-empty set and M an A-module . Then the set of maps Map(S , M) is an A-module . We have already noted previously that it is a commutative group , and for f E Map(S, M ) , a E A we define af to be the map such that (af>(s) = af(s ). The axioms for a module are then trivially verified. For further examples, see the end of this section.
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