Modules over Noncommutative Rings
This chapter contains two sets of tools for working with modules over a ring R with identity. The first set concerns finiteness conditions on modules, and the second set concerns the Hom and tensor product functors.
Sections 1–3 concern finiteness conditions on modules. Section 1 deals with simple and semisimple modules. A simple module over a ring is a nonzero unital module with no proper nonzero submodules, and a semisimple module is a module generated by simple modules. It is proved that semisimple modules are direct sums of simple modules and that any quotient or submodule of a semisimple module is semisimple. Section 2 establishes an analog for modules of the Jordan-Hölder Theorem for groups that was proved in Chapter IV; the theorem says that any two composition series have matching consecutive quotients, apart from the order in which they appear. Section 3 shows that a module has a composition series if and only if it satisfies both the ascending chain condition and the descending chain condition for its submodules.
Sections 4–6 concern the Hom and tensor product functors. Section 4 regards HomR(M, N), where M and N are unital left R modules, as a contravariant functor of the M variable and as a covariant functor of the N variable. The section examines the interaction of these functors with the direct sum and direct product functors, the relationship between Hom and matrices, the role of bimodules, and the use of Hom to change the underlying ring. Section 5 introduces the tensor product M ?R N of a unital right R module M and a unital left R module N, regarding tensor product as a covariant functor of either variable. The section examines the effect of interchanging M and N, the interaction of tensor product with direct sum, an associativity formula for triple tensor products, an associativity formula involving a mixture of Hom and tensor product, and the use of tensor product to change the underlying ring. Section 6 introduces the notions of a complex and an exact sequence in the category of all unital left R modules and in the category of all unital right R modules. It shows the extent to which the Hom and tensor product functors respect exactness for part of a short exact sequence, and it gives examples of how Hom and tensor product may fail to respect exactness completely.
KeywordsAbelian Group Tensor Product Exact Sequence Associative Algebra Simple Module
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