Let R be any ring (possibly without identity) and X a nonempty set. In this exercise an R-module Fis called a free module on X if F is a free object on X in the category of a/I left R-modules. Thus by Definition 1.7.7, F is the free module on X if there is a function _L: X → F such that for any left R-module A and function ∫: X → A there is a unique R-module homomorphism f̅: F → A with f̅_I, = f

(a) Let {X_i| i ϵ I) be a collection of mutually disjoint sets and for each i ϵ I, suppose F, is a free module on X_i with _Li : x_i → F_i, Let X = and F_i, with ϕ_i: F_i → F the canonical injection. Define _L : X → F by _i(x) = ϕ_iLi(x) for x ϵ X_i;(_L is well defined since the X_i are disjoint). Prove that F is a free module on X. 

(b) Assume R has an identity. Let the abelian group Z be given the trivial R-module structure (rm = 0 for all r ϵ R, m ϵ Z), so that R⊕z is an R-module with r(r' ,m) = (rr', 0) for all r ,r' ϵ R, m ϵ Z. If X is any one element set, X = { t) , let _L : X → R⊕Z be given by _L(t) = (l_R,1). Prove that R⊕Z is a free module on X.

(c) If R is an arbitrary ring and X is any set, then there exists a free module on X.

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U X₂ iel