General Proofs Krull-Schmidt and Jordan-Holder

Question

We have versions of the Krull-Schmidt and Jordan-Holder theorems for both groups and modules. Together, these take care of all our favourite algebraic structures, since rings, fields and vector spaces are modules. However, the module versions only prove the corresponding results for Abelian groups, since you can only have a module over an Abelian group. The results definitely don't generalise to algebras in the sense of universal algebra, so I was wondering if there were some condition that allowed us to prove the results in general. The obvious idea is to define "rings" over non-Abelian groups, "modules" over these "rings", and prove the natural analogues of the theorems for these "modules".

Answer 1

For a possibly nonempty set $X$ and a group $G$, we call $G$ an $X$-group if for each $x\in X$ and $g\in G$ there is an element $x\cdot g\in G$ such that $x\cdot (gh)=(x\cdot g)(x\cdot h)$.

The Jordan-Holder and Krull-Schmidt theorems can be proved for $X$-groups, and this is precisely what is done in

Isaacs, I. Martin. Algebra: a graduate course. Vol. 100. American Mathematical Soc., 2009.

That's really the only place I've seen it done that way! So you don't need groups over any sort of ring at all, you just need a set acting on a group.