(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let R be a commutative ring, and let [itex] F = R^{\oplus B} [/itex] be a free R-module over R. Let m be a maximal ideal of R and take [itex] k = R/m [/itex] to be the quotient field. Show that [itex] F/mF \cong k^{\oplus B} [/itex] as k-vector spaces.

3. The attempt at a solution

If we remove the F and k notations, we essentially just want to show that

[tex] R^{\oplus B}/ m R^{\oplus B} \cong (R/m)^{\oplus B} [/tex]

and so it seems like we should use the first isomorphism theorem.

Now we define [itex] R^{\oplus B} = \{ \alpha: B \to R, \alpha(x) = 0 \text{ cofinitely in } B \} [/itex]. If [itex] \pi : R \to R/m [/itex] is the natural projection map, define [itex] \phi: R^{\oplus B} \to (R/m)^{\oplus B} [/itex] by sending [itex] \alpha \mapsto \pi \circ \alpha [/itex]. This is an R-mod homomorphism, and is surjective by the surjectivity of the projection. Hence we need only show that the kernel of this map is given by [itex] mR^{\oplus B} [/itex]

Now

[tex]\begin{align*}

\ker\phi &= \{ \alpha:B \to R, \forall x \in B \quad \pi(\alpha(x)) = 0_k \} \\

&= \{ \alpha:B \to R, \forall x \in B, \alpha(x) \in m \} \end{align*}

[/tex]

where I may have skipped a few steps in this derivation, but I think this is right. Now it's easy to show that [itex] mF \subseteq \ker \phi [/itex] since m is an ideal. However, the other inclusion is where I'm having trouble.

I guess maybe the whole question could be rephrased to avoid the baggage that comes with the question. Namely, if [itex] \alpha: B \to R [/itex] is such that [itex] \forall x \in B, \alpha(x) [/itex] lies in a maximal ideal of R, should that [itex] \alpha = r \beta [/itex] for some [itex] \beta: B \to R [/itex].

Edit: I guess I'm hoping to show they're isomorphic as R-modules, and then push that over to k-vector spaces. Maybe this is where my mistake is coming from?

Edit 2: Fixed mistake made in "without baggage" statement.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Vector spaces as quotients of free modules

**Physics Forums | Science Articles, Homework Help, Discussion**