Semidistributive and distributively decomposable rings
type: Item
instance of: scholarly article
title: Semidistributive and distributively decomposable rings
zbMATH Open document ID: 0944.16004
DOI: 10.1007/BF02679824
published in: Mathematical Notes
publication date: 2000-09-18T00:00:00Z
review text: A module \(M_A\) is called semidistributive if it is a direct sum of distributive modules. A ring \(A\) is right distributively decomposable if its identity element is a sum of nonzero orthogonal idempotents \(e_1,\dots,e_n\) such that all \(e_iAe_i\) are right distributive rings. In the present note the right semidistributive rings, as well as distributively decomposable rings are investigated. If \(A\) is a Noetherian right distributively decomposable ring, then \(\bigcap_{i=1}^\infty J(A)^i=0\) and the right Krull dimension of \(A\) is less than or equal to 1. If \(A\) is a semiperfect semiprime right hereditary ring, then the following conditions are equivalent: (1) \(A\) is a right Noetherian ring; (2) \(A\) is a right serial ring; (3) \(A\) is a right semidistributive ring. Some results on the representation of the ring \(A\) as a finite direct product of right distributively decomposable rings are indicated.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1407525
zbMATH Keywords: direct sums of distributive modules
zbMATH Keywords: orthogonal idempotents
zbMATH Keywords: right distributive rings
zbMATH Keywords: right semidistributive rings
zbMATH Keywords: distributively decomposable rings
zbMATH Keywords: right Krull dimension
zbMATH Keywords: semiperfect semiprime right hereditary rings
zbMATH Keywords: right Noetherian rings
zbMATH Keywords: right serial rings
author: A. A. Tuganbaev
reviewed by: A. I. Kashu
MaRDI profile type: Publication
cites work: Direct sums of distributive modules
cites work: Q4891444
cites work: Krull dimension
cites work: Q3748429
cites work: Noetherian Rings with Krull Dimension One
cites work: Distributive semiprime rings
cites work: Modules Whose Lattice of Submodules is Distributive
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description: scientific article; zbMATH DE number 1407525