Hochschild (co)homologies of dg $K$-rings and their Koszul duals

We formulate the (co)bar construction theory of dg $K$-(co)rings and the calculus theory of the Hochschild homology and cohomology of dg $K$-rings. As applications, we compare the Hochschild (co)homologies of a complete typical dg $K$-ring and its Koszul dual. Moreover, we show that the Koszul dual of a finite dimensional complete typical $d$-symmetric dg $K$-ring is a $d$-Calabi-Yau dg algebra whose Hochschild cohomology is a Batalin-Vilkovisky algebra. Furthermore, we prove that the Hochschild cohomologies of a finite dimensional complete typical $d$-symmetric dg $K$-ring and its Koszul dual are isomorphic as Batalin-Vilkovisky algebras. In conclusion, we found a connection between the Batalin-Vilkovisky algebra structures on the Hochschild cohomologies of $d$-Calabi-Yau dg algebras and $d$-symmetric dg $K$-rings.