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 \(B \rightarrow A\) 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 \(G(B) = \bigoplus _{i \ge 0}\mathfrak {n} ^i/\mathfrak {n} ^{i+1}\) 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).
