Extensions of uniserial modules

2020 ◽  
Vol 144 ◽  
pp. 73-86
Author(s):  
Gabriella D'Este ◽  
Fatma Kaynarca ◽  
Derya Keskin Tütüncü
Keyword(s):  
2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.


2014 ◽  
Vol 14 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. Ghorbani ◽  
Z. Nazemian

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.


1993 ◽  
Vol 21 (5) ◽  
pp. 1645-1656 ◽  
Author(s):  
John Fitzgerald
Keyword(s):  

1997 ◽  
Vol 193 (1) ◽  
pp. 102-121 ◽  
Author(s):  
Nguyen Viet Dung ◽  
Alberto Facchini

1989 ◽  
Vol 17 (6) ◽  
pp. 1463-1476 ◽  
Author(s):  
Jusuf Alajbegović ◽  
Paolo Zanardo
Keyword(s):  

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.


1993 ◽  
Vol 86 (1) ◽  
pp. 35-50
Author(s):  
P.C. Eklof ◽  
S. Shelah
Keyword(s):  

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