Generalization of Vector spaces

Question

The Definition of Vector Spaces' operations seems awfully close to Algebraic operations on a Ring. I'm sure there probably is a Generalized version of a Vector space definition using an arbitrary Ring instead of the usual 'additive' $+$ or Scalar ( from $\mathbb{R}$ or $\mathbb{C}$ ) multiplicative ($\lambda$) to a more general Group/Ring Action on a Set? The advantage is then we could study say use the Fundamental Group $\pi(1)$ ( From Algebraic topology) action on Arbitrary sets. Vector Spaces being instances of such spaces as applied to Familiar $\mathbb{R}$ etc. Thanks