When a group ring is linearly compact or algebraically compact
A group ring $R[G]$ is right linearly compact (resp. right algebraically compact) if and only if $R$ is right linearly compact (resp right algebraically compact) and $|G|$ is finite.
Reference(s)
W. Zimmermann. ($\Sigma$-) algebraic compactness of rings. (1982) @ Theorem 1 and Corollary 2, p 320