Abstract : Let $D$ be an integrally closed domain with quotient field $K$, $*$ be a star operation on $D$, $X, Y$ be indeterminates over $D$, $N_*$ = $\{f \in D[X]|$ $(c_D(f))^*$ = $D\}$ and $R = D[X]_{N_*}$. Let $b$ be the $b$-operation on $R$, and let $*_c$ be the star operation on $D$ defined by $I^{*_c} = (ID[X]_{N_*})^b \cap K$. Finally, let $Kr(R,b)$ (resp., $Kr(D,*_c)$) be the Kronecker function ring of $R$ (resp., $D$) with respect to $Y$ (resp., $X,Y$). In this paper, we show that $Kr(R,b) \subseteq Kr(D,*_c)$ and $Kr(R,b)$ is a kfr with respect to $K(Y)$ and $X$ in the notion of [2]. We also prove that $Kr(R,b) = Kr(D,*_c)$ if and only if $D$ is a P$*$MD. As a corollary, we have that if $D$ is not a P$*$MD, then $Kr(R,b)$ is an example of a kfr with respect to $K(Y)$ and $X$ but not a Kronecker function ring with respect to $K(Y)$ and $X$.

Keywords : ($e.a.b.$) star operation, Kronecker function ring (KFR), kfr, Nagata ring, P$*$MD