Math 762 Spring 2016 Homework 1 Drew Armstrong
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چکیده
Problem 1. Infinite Products and Coproducts in Ab. We have seen that finite products and coproducts agree in Ab. However, the same is not true for infinite products and coproducts. Let I be a set and let {Ai}i∈I be a family of abelian groups, each equal to some fixed group A. (a) Show that the set AI := HomSet(I, A) is an abelian group. Furthermore, show that we can think of this group as the infinite product Πi∈IAi in the category Ab. (b) Let A⊕I denote the subgroup of AI in which all but finitely many elements of I are sent to the identity element 0 ∈ A. Show that we can think of A⊕I as the infinite coproduct ⊕ i∈I Ai in the category Ab. (c) Show that the inclusion A⊕I ⊆ AI can be strict. [Hint: Let A = Z/10Z and I = Z.]
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