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Abstract Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V , we study the relationship between various types of V -modules and modules for the higher level Zhu algebras for V , denoted A n (V) , for n ∈ N , first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not A n − 1 (V) is isomorphic to a direct summand of A n (V) affects the types of indecomposable V -modules which can be constructed by inducing from an A n (V) -module, and in particular whether there are V -modules induced from A n (V) -modules that were not already induced by A 0 (V). We give some characterizations of the V -modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of A 1 (V) : when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of A 1 (V) in relationship to A 0 (V) determines what types of indecomposable V -modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules. [ABSTRACT FROM AUTHOR]

Abstract Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R -module is an R -module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain. [ABSTRACT FROM AUTHOR]