Chapter X. Modules over Noncommutative Rings

Abstract

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 $\operatorname{Hom}$ 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 $\operatorname{Hom}$ and tensor product functors. Section 4 regards $\operatorname{Hom}_R(M,N)$, 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 $\operatorname{Hom}$ and matrices, the role of bimodules, and the use of $\operatorname{Hom}$ to change the underlying ring. Section 5 introduces the tensor product $M\otimes_RN$ 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 $\operatorname{Hom}$ 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 $\operatorname{Hom}$ and tensor product functors respect exactness for part of a short exact sequence, and it gives examples of how $\operatorname{Hom}$ and tensor product may fail to respect exactness completely.