Rings of Teter type

Let $R$ be a Cohen--Macaulay local $K$-algebra or a standard graded $K$-algebra over a field $K$ with a canonical module $\omega_R$. The trace of $\omega_R$ is the ideal $tr(\omega_R)$ of $R$ which is the sum of those ideals $\varphi(\omega_R)$ with $\varphi\in Hom_R(\omega_R,R)$. The smallest number $s$ for which there exist $\varphi_1, \ldots, \varphi_s \in Hom_R(\omega_R,R)$ with $tr(\omega_R)=\varphi_1(\omega_R) + \cdots + \varphi_s(\omega_R)$ is called the Teter number of $R$. We say that $R$ is of Teter type if $s = 1$. It is shown that $R$ is not of Teter type if $R$ is generically Gorenstein. In the present paper, we focus especially on $0$-dimensional graded and monomial $K$-algebras and present various classes of such algebras which are of Teter type.