Generalizations of Nakayama ring. III
type: Item
instance of: scholarly article
title: Generalizations of Nakayama ring. III
zbMATH Open document ID: 0612.16010
author: Manabu Harada
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: [For part II see the review above.] Using the same definitions and notations as in the above paper, the author continues his study of right Artinian rings having (*,n) or (**,n). In the first section a right Artinian ring with (*,1) for any hollow module is discussed. It is observed that if \(J^ 2=0\), then R satisfies (*,2) and hence (*,1). If R is an algebra over an algebraically closed field \(K_ 0\), then R satisfies II'': \(eRe/eJe=\bar eK'\) for each primitive idempotent e, K' a field in the center of R. It is shown that for an algebra R over a field K with condition II'', the following are equivalent: 1) R satisfies (*,2) and \(eJ=A_ 1\oplus B_ 1\) such that \(A_ 1/J(A_ 1)\not\approx B_ 1/J(B_ 1)\), \(A_ 1\), \(B_ 1\) hollow. 2) R satisfies (*,n) for all n. In the above paper US-3 rings with (*,1) or (*,2) were studied. Now US-4 algebras over an algebraically closed field are observed. Provided that \(J^ 3=0\) \((J^ 2\neq 0)\) all lattices of submodules in eR are given for each primitive idempotent e. In the final section R is an algebra over a field with condition II''. Necessary and sufficient conditions are given in order that R be a US-4 algebra with (*,1) in terms of the structure of eR (e a primitive idempotent). With a slight modification a similar structure theorem is obtained for (*,2).
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 3989585
zbMATH Keywords: right Artinian rings
zbMATH Keywords: hollow module
zbMATH Keywords: primitive idempotent
zbMATH Keywords: US-3 rings
zbMATH Keywords: US-4 algebras
zbMATH Keywords: lattices of submodules
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3989585