Typically, a free algebraic object is one that can generated from a set without any relations (quotients) applied. When it's a free module, each element of that set is used as a basis …

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Jul 31, 2018 · Here I asked a question about the tensor products- Construction of tensor product over module and apparently I do not understand what a free module is. More specifically, in …

Feb 15, 2013 · The basis of a free module need not be finite, so free does not imply finitely generated. You are correct that for a general finitely generated module, there may be relations …

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, …

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Nov 20, 2012 · In Artin's Algebra he says that $R^{n}$, where $R$ is a ring, is a free $R$ module. But if $R$ contains zero divisors then how can $R$ be free since there exists $r,r ...

$\begingroup$ If $\Bbb{Q}$ was free, it must have a basis (I assume you mean free abelian (free as a $\Bbb{Z}$-module)). Show the rationals aren't cyclic (so not of rank 1), and that any two …

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Oct 22, 2011 · A submodule of a finitely generated free module over a principal ideal domain is free. (For example, we can use the structure theorem for finitely generated modules over …