Background and motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1_\mathbb{Z}(A,\mathbb{Z})$. If $A$ is free then this abelian group is trivial. Is the converse true? The converse is known as the Whitehead problem.
Now the Whitehead problem was shown independent of ZFC by Shelah. This is somewhat unsatisfying, but it can be proved assuming the axiom of constructibility ($V=L$). In fact, there are other reasonable sounding and simple statements in analysis and topology that are also independent from ZFC but become theorems once $V=L$ is assumed. Another statement is that the global (a.k.a. homological) dimension of the ring $\prod_{i=1}^\infty \mathbb{C}$ is two if and only if the continuum hypothesis holds, which is implied by adding $V=L$ again. Thus I feel warmly about the axiom of constructibility.
Of course, the dissatisfaction remains, because there are other statements that are independent from $ZFC + V=L$ (well, perhaps I should write $ZF + V=L$ to save space). Question 18058 and Question 11480 are examples.
Question, loosely stated
Now, I am curious if there are any known algebraic (see postscript) statements, reasonably naturally sounding (use judgement), that are independent from $ZF + V=L$? Or perhaps independent from $ZF + A$ where $A$ is your favourite set-theoretic axiom independent from $ZFC$? Perhaps some easy low-hanging fruit for this search would be in the area of homological dimension theory? Has anyone done work on this type of thing?
I am sure there must be some statements of some kind. In the proof of the Whitehead problem under adding $V=L$, one can first deduce some combinatorial statement that requires little set-theoretic machinery and then use it to prove the Whitehead problem. So perhaps adding other axioms, one can also deduce various combinatorial gadgets and them use them to get new algebraic statements that are independent from the original $ZFC$? I would even like to hear about statements implied by additional axioms, but whose independence is not proved. (One of Devlin's books explains this).
(Remark: Although with enough brute force, one should be able to churn out such things no matter how many new axioms one adds, I would be interested in finding enough axioms of set theory so that the remaining independent statements would be so bizarre sounding that they would be essentially be uninteresting for all of time. Presumably as one adds more and more axioms to set theory, this would happen, no?)
Since I am not a set theorist, I would appreciate answers that are understandable to someone who knows the basics of set theory (say a typical first grad course) but knows very little about forcing.
Postscript
By "algebraic" I mean roughly some statement in the language of groups, rings, ideals, modules, fields, etc., somewhat natural sounding, that does not itself refer to the additional axiom (e.g. using some set whose cardinal is inaccessible, or something along these lines). Thus I am not looking for statements about real numbers or set theory, although those are interesting too.
