On pure Goldie dimensions

In this paper, we examine the pure Goldie dimension and dual pure Goldie dimension in finitely accessible additive categories. In particular, we show that if A is an object in a finitely accessible additive category 𝒜 that has finite pure Goldie dimension n and finite dual pure Goldie dimension m, then End_𝒜(A) is semilocal and the dual Goldie dimension of End_𝒜(A) is less than or equal to n+m.     

A Note on Dimension Modules

In [2 Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 .[Crossref], [Web of Science ®] , [Google Scholar]] Camillo and Zelmanowitz stated that rings all whose modules are dimension modules are semisimple Artinian. It seem however that the proof in [2 Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 .[Crossref], [Web of Science ®] , [Google Scholar]] contains a gap and applies to rings with finite Goldie dimension only. In this paper we show that the result indeed holds for all rings with a basis as well as for all commutative rings with Goldie dimension attained.     

有有限Goldie维数模的直和消去

In this paper, we show that if C is a module with finite Goldie dimension, and A ＋ C≌B ＋ C, then there are essential submodules A＇△ A, B＇△ B such that A＇≌ B＇.     

关于SF－环的几点注记

本文中，我们证明了如下主要结果：Ⅰ　对于环Ｒ，下面条件是等价的：（１）Ｒ是Ａｒｔｉｎ半单环；（２）Ｒ是左ＳＦ－环，且Ｒ满足特殊右零化于降链条件；（３）Ｒ是左ＳＦ－环和Ｉ－环，且Ｒ￣Ｒ具有有限Ｇｏｌｄｉｅ维数。Ⅱ对于环Ｒ，下面条件是等价的：（１）Ｒ是ＶｏｎＮｅｕｍａｎｎ正则环；（２）Ｒ是左ＳＦ－环，且每个苛异循环左Ｒ－模的极大子模是平坦的。     

The Osofsky–Smith Theorem for Modular Lattices and Applications (I)

The aim of this article is to present a latticial version of the renown module theoretical Osofsky–Smith Theorem.