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.

Printable View

- Dec 4th 2011, 09:02 AMjcir2826Modules
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. - Dec 4th 2011, 09:34 AMDrexel28Re: Modules
Consider the free module $\displaystyle R^{\oplus X}$ (i.e. $\displaystyle X$-fold coproduct). This has a natural basis of the form $\displaystyle \{e_x:x\in X\}$ where $\displaystyle e_x$ is the tuple with $\displaystyle 1$ in the $\displaystyle x^{\text{th}}$ coordinate and zero elsewhere. I think the rest should be obvious.

- Dec 4th 2011, 07:54 PMjcir2826Re: Modules
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?

- Dec 4th 2011, 08:19 PMDrexel28Re: Modules