The purpose of this paper is to introduce some new classes of rings and modules that are closely related to the classes of almost Dedekind domains and almost Dedekind modules. We introduce the concepts of $\phi$-almost Dedekind rings and $\Phi$-almost Dedekind modules and study some properties of this classes. In this paper we get some equivalent conditions for $\phi$-almost Dedekind rings and ...

Invertibility of multiplication modules All rings are commutative with 1 and all modules are unital. Let R be a ring and M an R-module. M is called multiplication if for each submodule N of M, N=IM for some ideal I of R. Multiplication modules have recently received considerable attention during the last twenty years. In this talk we give the de nition of invertible submodules as a natural gene...

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an א0-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring with a “nice” topology is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite excha...

Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm–Euclidean. Let R be the ring of integers of an algebraic number field ...

For a closure space (P,φ) with φ(∅) = ∅, the closures of open subsets of P , called the regular closed subsets, form an ortholattice Reg(P,φ), extending the poset Clop(P,φ) of all clopen subsets. If (P,φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, ...

We discuss a trinity, i.e., three distinct expressions, of the Borcherds Φ-function on the analogy of the trinity of the Dedekind η-function.