Direct Sums and Products
For each ring R we have derived several module categories—among these the category R M of left R-modules. This derivation is not entirely reversible for, in general, R M does not characterize R. However, as we shall see in Chapter 6 it does come close. Thus, we can expect to uncover substantial information about R by mining R M. So in this chapter we start to probe more deeply into the structure of the modules themselves. In so far as possible we propose to do this in the context of the category R M for in this way at any subsequent stage we shall be able to apply the general machinery of category theory.
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