In this paper, we compute the projective class ring of the new type restricted quantum group ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$. First, we describe the principal indecomposable projective ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$-modules and study their radicals, composition series, Cartan matrix of ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$ and so on. Then, we deconstruct the tensor products between two simple modules, two indecomposable projective modules and a simple module and an indecomposable projective module, into direct sum of some indecomposable representations. At last, we characterize the projective class ring by generators and relations explicitly.

Let $A$ be a finite-dimensional algebra with identity over the field $\mathbb{F}$, $U(A)$ be the group of units of $A$ and $L(A)$ be the set of left ideals of $A$. It is well known that there is an equivalence relation $\sim$ on $L(A)$ by defining $L_1\sim L_2\in L(A)$ if and only if there exists some $u\in U(A)$ such that $L_1=L_{2}u$. $C(A)=\{[L]|L\in L(A)\}$ is the set of equivalence classes determined by the relation $\sim$, which is a semigroup with respect to the natural operation $[L_1][L_2]=[L_1{L}_2]$ for any $L_1,L_2\in L(A)$. The purpose of this paper is to describe the structures of semigroup of conjugacy classes of left ideals for the Sweedler's four-dimensional Hopf algebra $H_4$.