We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R = Z, and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, respectively. More generally, whenever R is a Dedekind domain, we extend the...

Let $R$ be a commutative ring with nonzero identity. An $R$-module $M$ is called $\phi$-P-flat if $x \in \Ann(s)M$ for every non-nilpotent element $s R$ and $x\in M$ such that $sx=0$. In this paper, we introduce study the class of $\phi$-PF-rings, i.e., rings in which all ideals are $\phi$-P-flat. Among other results, transfer $\phi$-PF-ring to amalgamation investigated. Several examples deline...

The main theorem of this paper complements the tame-wild dichotomy for commutative Noetherian rings, obtained by Klingler and Levy [14]–[16]. They gave a complete classification of all finitely generated modules over Dedekind-like rings (cf. Definition 1.1) and showed that, over any ring that is not a homomorphic image of a Dedekind-like ring, the category of finite-length modules has wild repr...

for any given finite abelian group‎, ‎we give factorizations of the group determinant in the group algebra of any subgroups‎. ‎the factorizations is an extension of dedekind's theorem‎. ‎the extension leads to a generalization of dedekind's theorem‎.

Abstract. Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent poly...

Let R be a Dedekind domain. In [6], Enochs’ solution of the Flat Cover Conjecture was extended as follows: (∗) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set. We show that (∗) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP implies that C is not cogenerated by a set whenever C is a cotorsi...

In these notes we will first define projective modules and prove some standard properties of those modules. Then we will classify finitely generated projective modules over Dedekind domains Remark 0.1. All rings will be commutative with 1. 1. Projective modules Definition 1.1. Let R be a ring and let M be an R-module. Then M is called projective if for all surjections p : N → N ′ and a map f : ...

We show that cohopfian modules with finite exchange have countable exchange. In particular, a module whose endomorphism ring is Dedekind-finite and π-regular has the countable exchange property. We also show that a module whose en-domorphism ring is Dedekind-finite and regular has full exchange. Finally, working modulo the Jacobson radical, we prove that any module with the (C 2) property and a...

We call a theory a Dedekind theory if every complete quantifier-free type with one free variable either has a trivial positive part or it is isolated by a positive quantifier-free formula. The theory of vector spaces and the theory fields are examples. We prove that in a Dedekind theory all positive quantifier-free types are principal so, in a sense, Dedekind theories are Noetherian. We show th...