Reducing a set of generators to a free base

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Given an algebraic structure $ A $ which is free on some subset of its underlying set, does every generating subset of $ A $ contain a free generating set?

For vector spaces, this is true, what about the general case?

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The free abelian group $\;\Bbb Z\;$ is generated by $\;\{2,3\}\;$ . Can you find a free generating subset in this case?