A Note on Generating a Power Basis over a Dedekind Ring

In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

Relationship of Essentially Small Quasi-Dedekind Modules with Scalar and Multiplication Modules

Let be a ring with 1 and D is a left module over . In this paper, we study the relationship between essentially small quasi-Dedekind modules with scalar and multiplication modules. We show that if D is a scalar small quasi-prime -module, thus D is an essentially small quasi-Dedekind -module. We also show that if D is a faithful multiplication -module, then D is an essentially small prime -module iff is an essentially small quasi-Dedekind ring.

On Syzygy Modules over Laurent Polynomial Rings

In this paper, we present a dynamical method for computing the syzygy module of multivariate Laurent polynomials with coefficients in a Dedekind ring (with zero divisors) by reducing the computation over Laurent polynomial rings to calculations over a polynomial ring via an appropriate isomorphism.

Strongly graded rings which are generalized Dedekind rings

Let [Formula: see text] be a strongly graded ring of type [Formula: see text] such that [Formula: see text] is a prime Goldie ring with its quotient ring [Formula: see text]. It is shown that the following three conditions are equivalent: (i) [Formula: see text] is a [Formula: see text]-invariant generalized Dedekind ring ([Formula: see text]-Dedekind ring for short), (ii) [Formula: see text] is a [Formula: see text]-Dedekind ring and (iii) [Formula: see text] is a graded [Formula: see text]-Dedekind ring. We describe all invertible ideals of [Formula: see text]-Dedekind rings in terms of [Formula: see text] and [Formula: see text]. We provide counterexamples of [Formula: see text]-invariant [Formula: see text]-Dedekind rings which are not [Formula: see text]-Dedekind rings.