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# A First Course of Homological Algebra by D. G. Northcott By D. G. Northcott

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Applying the functor Hom z (-, Q) to this sequence we obtain an exact sequence 0-*A**-+F* in <€\ and, by Theorem 13, F* is A-injective. Let aeA. e. for a l l / i n Homz(A, Q). Clearly ^ e H o m z ( i * , Q) = ^ * * . We therefore have a mapping 0:^4^,4** in which (a) = a. Next is additive since ai+a2 = ^ a i + ^ a 2 . If AeA, then (A0a) (/) = ^O(/A) = (/A) (a). But (/A)(a)=/(Aa). Consequently (A^a) (/) =f(Aa) = ^ Ao (/) and therefore A^a = ^ A a . This shows that 0:^4 ^^4** is a homomorphism of left A-modules.

Conversely, suppose E is divisible and let / be any non-zero ideal of Z. Consider a diagram E SOLUTIONS TO EXERCISES 49 in ^ z . We have / = Zm for some non-zero integer m. Put e — f(m). Since mE = E there is an element e' in E such that e = me'. Define a homomorphism h:Z-+E by h(n) = ne'. Then h extends/. That E is injective now follows from Theorem 11. Exercise 14. Show that every left A-module is protective if and only if every left A-module is injective. Solution. First suppose that every left A-module is injective and that A is a left A-module.

Then C is exact. It remains for us to show that i/r is surjective. Suppose that gi:G->D(i= 1,2) are A-homomorphisms such that g^ = g2ifr = 0 (say). Then 6