Higher dimensional Teter rings

Let $(A,\mathfrak{m})$ be a complete Cohen-Macaulay local ring. Assume $A$ is not Gorenstein. We say $A$ is a Teter ring if there exists a complete Gorenstein ring $(B,\mathfrak{n})$ with $\dim B=\dim A$ and a surjective map $B\to A$ with $e(B)-e(A)=1$ (here $e(A)$ denotes multiplicity of $A$). We give an intrinsic characterization of Teter rings which are domains. We say a Teter ring is a strongly Teter ring if $G(B)={⨁}_{i\ge 0}{\mathfrak{n}}^i\mathrm{/}{\mathfrak{n}}^{i+1}$ is also a Gorenstein ring. We give an intrinsic characterizations of strongly Teter rings which are domains. If $k$ is algebraically closed field of characteristic zero and $R$ is a standard graded Cohen-Macaulay $k$-algebra of finite representation type (and not Gorenstein) then we show that $\widehat{R_{\mathfrak{M}}}$ is a Teter ring (here $\mathfrak{M}$ is the maximal homogeneous ideal of $R$).