Algebraic actions are rich sources of examples in dynamics.They have been studied extensively when the acting groups are $$extract_itex$$\mathbb{Z}^d$$/extract_itex$$ by the works of Doug Lind, Klaus Schmidt, Thomas Ward,... The fact that the integral group ring of $$extract_itex$$\mathbb{Z}^d$$/extract_itex$$ is a commutative factorial Noetherian ring plays a vital role for such study, as it makes the machinery of commutative algebra available. In this talk we will present recent results joint with Hanfeng Li and Andreas Thom for algebraic actions of general countable groups. Operator algebras, especially group von Neumann algebras and $$extract_itex$$\ell^1$$/extract_itex$$-algebras play important roles in our work."