A j}, j E J , there exists a 6. Dualization, Injective Modules 29 unique homomorphism cp: M - T such that for every j T. 3). We therefore say that the notion of the direct product is dual to the notion of the direct sum. ). This can be done for a great many - though not all - of the basic notions introduced in Sections 1, ... ,5.

LR may be written uniquely in the form (b, A) where bEP(j) and AEA j. lR-+A by fj(a) = A. lR-imfj . Clearly imfj is an ideal in A. Since A is a principal ideal domain, this ideal is generated by one element, say Aj . lR, such that fj(c) = Aj . Let J' ~ J consist of those j such that Aj =1= O. We claim that the family {c j }, j E J', is a basis of R. n First we show that {cj},j E J', is linearly independent. Let L Il k h = 0 C. k=1 and let jl

Show that if a category has a zero object, then every initial object, and every terminal object, is isomorphic to that zero object. Deduce that the category of sets has no zero object. 2. Functors Within a category <£ we have the morphism sets <£(X, Y) which serve to establish connections between different objects of the category. Now the language of categories has been developed to delineate the various areas of mathematical theory; thus it is natural that we should wish to be able to describe connections between different categories.