Local Dedekind Domain Research Articles

Classification of 1-absorbing comultiplication modules over a pullback ring

One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

Quasi semiprime multiplication modules over a pullback of a pair of Dedekind domains

The main purpose of this article is to classify all indecomposable quasi semiprime multiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi semiprime multiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi semiprime multiplication modules and classify quasi semiprime multiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi semiprime multiplication modules and then, using this list of separated quasi-semiprime multiplication modules, we show that non-separated indecomposable quasi semiprime multiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi semiprime multiplication modules.

On a new variation of injective modules

In this paper, we provide various properties of GE and GEE-modules, a new variation of injective modules. We call M a GE-module if it has a g-supplement in every extension N and, we call also M a GEE-module if it has ample g-supplements in every extension N. In particular, we prove that every semisimple module is a GE-module. We show that a module M is a GEE-module if and only if every submodule is a GE-module. We study the structure of GE and GEE-modules over Dedekind domains. Over Dedekind domains the class of GE-modules lies between WS-coinjective modules and Zoschinger's modules with the property (E). We also prove that, if a ring R is a local Dedekind domain, an R-module M is a GE-module if and only if M≅(R^{∗})ⁿ⊕K⊕N, where R^{∗} is the completion of R, K is injective and N is a bounded module.

Generalizations of ⊕ -supplemented modules

We introduce ⊕ -radical supplemented modules and strongly ⊕ -radical supplemented modules (briefly, srs ⊕-modules) as proper generalizations of ⊕ -supplemented modules. We prove that (1) a semilocal ring R is left perfect if and only if every left R-module is an ⊕ -radical supplemented module; (2) a commutative ring R is an Artinian principal ideal ring if and only if every left R-module is an srs ⊕-module; (3) over a local Dedekind domain, every ⊕ -radical supplemented module is an srs ⊕-module. Moreover, we completely determine the structure of these modules over local Dedekind domains.

A variation of supplemented modules

Over a general ring, an R-module is w-supplemented if and only if amply w-supplemented. It is proved that over a local Dedekind domain, all modules are w-supplemented and over a non-local Dedekind domain, an R-module M is w-supplemented if and only if Soc(M)\ll M or M=S_0 \oplus (\bigoplus_{i \in I} K), where S_0 is a torsion, semisimple submodule of M and K is the field of quotients of R.

Strongly radical supplemented modules

Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.

Comultiplication modules over a pullback of Dedekind domains

First, we give complete description of the comultiplication modules over a Dedekind domain. Second, if R is the pullback of two local Dedekind domains, then we classify all indecomposable comultiplication R-modules and establish a connection between the comultiplication modules and the pure-injective modules over such domains.

On Prime Modules over Pullback Rings

First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if R is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime R-modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.

Ramification of valuations

In this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields. In characteristic zero and arbitrary dimension, we obtain the strongest possible generalization of the classical ramification theory of local Dedekind domains. We further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic. We prove that local monomialization and simultaneous resolution hold under very mild assumptions, and give pathological examples.

ON SECONDARY MODULES OVER PULLBACK RINGS

ABSTRACT Let R be the pullback, in the sense of[5], of two local Dedekind domains. We classify all those indecomposable representable R -modules M with finite-dimensional top, that is, such that is finite dimensional over and to establish a connection between the secondary modules and the pure-injective modules over such rings.

On Secondary Modules over Dedekind Domains

The aim of this paper is to classify the indecomposable secondary modules over local dedekind domains and to establish a connection between the indecomposable secondary modules and the indecomposable pure-injective modules over such domains.

Minimal injective resolution of an order over a local dedekind domain

(1998). Minimal injective resolution of an order over a local dedekind domain. Communications in Algebra: Vol. 26, No. 2, pp. 447-451.

A criterion for isomorphic tiled orders over a local Dedekind domain

Throughout this note R will denote a local Dedekind domain with a unique maximal ideal zR and the quotient ring K. Let n(^2) be an integer. As usual, an i?-order A in the full nXn matrix ring (K)n over K is an i?-subalgebra of (K)n such that A is finitely generated as an i?-module and AK=(K)n. An i?-orderin (K)n is called tiledif it contains n orthogonal idempotents. If A is a tiled Z?-order in (K)n then up to conjugation we may assume that it contains the idempotents en (l^z^n), where {etj\l which is constructed from their graph theoretic properties. In this note we shall remove the hypothesis between (xR)n and (/?), and prove the following

On local orders

Let R be a complete local Dedekind domain with quotient field K and let Λ be a local R-order in a separable K-algebra. This paper has two major goals. The first is to describe the local orders such that every uniform R-torsion-free module is tame. The second is to describe the pullback structure of local orders with finite representation type and such that every uniform R-torsion-free module is tame. We acomplish these goals by constructing an “ n-ad” order which is, in a certain sense, a maximal local order. This n-ad order provides new characterizations for finite type.

Tiled Orders of Finite Global Dimension

We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let $D$ be a local Dedekind domain with the quotient ring $K$. We show that for $2 \leq n \leq 5$, every tiled $D$-order of finite global dimension in ${(K)_n}$ is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For $n \geq 6$, we give a tiled $D$-order in ${(K)_n}$ without this property, which is also a counterexample to Tarsy’s conjecture, saying that the maximum finite global dimension of such an order is $n - 1$.

On Gorenstein orders

Let R be a local Dedekind domain with quotient field K and let Λ be a R-order in a separable K-algebra A. This paper considers those orders Λ that are Gorenstein and that can be written as pullback of R-torsionfree Λ-modules. If the algebra A has nontrivial central idempotents, then we give a characterization for Gorenstein orders. Further, if R is complete and Λ is a local order with finite representation type such that every uniform R-torsionfree Λ-module is tame, then the pullback structure of a Gorenstein order can also be explicitly described.

Tiled orders of finite global dimension

We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let D D be a local Dedekind domain with the quotient ring K K . We show that for 2 ≤ n ≤ 5 2 \leq n \leq 5 , every tiled D D -order of finite global dimension in ( K ) n {(K)_n} is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For n ≥ 6 n \geq 6 , we give a tiled D D -order in ( K ) n {(K)_n} without this property, which is also a counterexample to Tarsy’s conjecture, saying that the maximum finite global dimension of such an order is n − 1 n - 1 .

Local orders whose lattices are direct sums of ideals

LetRRbe a complete local Dedekind domain with quotient fieldKKand letΛ\Lambdabe a localRR-order in a separableKK-algebra. This paper classifies those ordersΛ\Lambdasuch that every indecomposableRR-torsionfreeΛ\Lambda-module is isomorphic to an ideal ofΛ\Lambda. These results extend to the noncommutative case some results for commutative rings found jointly by this author and L. Levy.

Almost M-projectives and nakayama rings

In 1960, H. Bass [3] defined the notion of semi-perfect rings and, in 1963, E. Mares [l l] generalized it to that of semi-perfect modules. Since that time various kinds of further generalizations have been considered. In particular, K. Oshiro [ 161 studied quasi-semiperfect modules in connection with several important conditions on direct summands. The present paper is closely related to his paper as well as to the papers [ 12, 151 by Mueller and others and is cncerned with the condition (D,) (which is (C,) in [16]) over modules with direct decomposition into hollow modules. This is indeed motivated by the theorem [ 16, Theorem 3.51 that every quasisemiperfect module has the above type of decomposition. In Section 2 we shall clarify relationships between M-projectives over modules with direct decomposition into hollow modules and the condition (D,), which is dual to Cl.5, Theorem 81. In order to translate (D, ) in terms of homomorphisms, we shall introduce a new concept of almost A4-projectives to find some relations between almost M-projectives and a weaker condition (D’,). After preparing several basic results in Section 3, we shall, in Section 4, restrict ourselves to semi-perfect rings and give a series of characterizations of right Nakayama rings with special properties by making use of the concept of almost M-projectives. Finally, we shall classify modules with (D,) over a local Dedekind domain in Section 5.

The construction of maximal orders over a Dedekind domain

Suppose R is a complete local Dedekind domain with quotient field F, and let f(x) be a monic polynomial in R[x] having non-zero discriminant. We present here a new algorithm to construct the maximal order of the algebra Af = F[x]/f(x)F[x]. The new algorithm incorporates ideas of Zassenhaus (1975, 1980) concerning P-adic stability and the algebraic decomposition of A s . We show that it is always possible either