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  3. let d be a division ring, and let m be a right d-m

Let D be a division ring, and let M be a right D-module. Recall that a...

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Advance Math

Let D be a division ring, and let M be a right D-module. Recallthat a subset S ⊂ M is linearly independent (with respect to D) iffor any finite subset T ⊂ S, and elements at ∈ D for t ∈T, if sum of tat = 0, then all the at =0.

(a) If S ⊂ M is linearly independent, show that there exists amaximal linearly independent subset U of M that contains S, andthat U is a basis for M (that is,M is a free D-module).

(b) Suppose that S is a generating set (that is, for everyelement m ∈ M, there exists a finite subset T ⊂ S and at∈ D such that m = sum of tat). Show that there exists asubset U ⊂ S that is a basis for M.

(c)* Bonus for proving that all bases of M have the samecardinality, if it has a finite basis. (It is also true forinfinite bases.)

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