Thread: Modules

Given a set X, prove that there exists a free R-module F with a basis B for which there is a
bijection ϕ : B → X.

Originally Posted by jcir2826

Given a set X, prove that there exists a free R-module F with a basis B for which there is a
bijection ϕ : B → X.

Consider the free module (i.e. -fold coproduct). This has a natural basis of the form where is the tuple with in the coordinate and zero elsewhere. I think the rest should be obvious.

So I am just missing to show that there is a bijection from this natural basis to the free module direct summand R and X?

Originally Posted by jcir2826

So I am just missing to show that there is a bijection from this natural basis to the free module direct summand R and X?

The basis is and isn't a bijection?