This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{ Z}/p^s\mathbb{ Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For abelian $p$-group rings over $\widehat{ \mathbb{ Z}}_p$, we extend the isomorphism between reduced continuous $K_2$ and the first cyclic homology group to all finite abelian $p$-groups. A constructive proof using a generalized Artin-Hasse map yields an explicit splitting. This isomorphism is realized by extending Oliver's $p$-adic logarithm. We also characterize the map from reduced continuous $K_2$ to reduced linearized $K_2$, clarifying the links between $K_2$, cyclic homology, and K\"{ a}hler differentials.
Categories: math.KT