I was wondering about the following.
Let $A$ be an abelian group, $a_i$ variables indexed with some arbitrary set $I$ and assume we have an infinite set $E$ of linear equations in finitely many variables of the form $$n_1a_1 + \ldots + n_ka_k = b$$ with $n_i \in \mathbb Z, b \in A$.
If this system has no solution, then is there already a finite subset of equations in $E$ which are inconsistent? So if any finite subset of equations is solvable, is $E$ solvable?
This is reminiscent of the compactness theorem in 1st order logic, but of course you can't apply this for a concrete group. Am I missing something? I assume that this has an easy proof or a counterexample. What happens if we presume particularly nice groups (divisible, free) or vector spaces? Then this question is just about linear systems of equations.
Thanks for giving some insight