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

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

(a) If S ⊂ M is linearly independent, show that there exists a maximal linearly independent subset U of M that contains S, and that 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 every element m ∈ M, there exists a finite subset T ⊂ S and a_t ∈ D such that m = sum of ta_t). Show that there exists a subset U ⊂ S that is a basis for M.

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

Expert Solution

Colby Messinger answered 1 year ago

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

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