Extensions of uniserial modules

2020 ◽  
Vol 144 ◽  
pp. 73-86
Author(s):  
Gabriella D'Este ◽  
Fatma Kaynarca ◽  
Derya Keskin Tütüncü
Keyword(s):  
Uniserial Modules

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Prime Submodules ◽  
Uniserial Modules ◽  
Structure Formula

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?”.

On commutative rings with uniserial dimension

2014 ◽  
Vol 14 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. Ghorbani ◽  
Z. Nazemian
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Artinian Rings ◽  
Uniform Dimension ◽  
Uniserial Modules

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.

Uniserial modules forB2

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

scholarly journals Weak Krull–Schmidt for Infinite Direct Sums of Uniserial Modules

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

Uniserial modules over manis rings

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

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

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.

scholarly journals Explicitly non-standard uniserial modules

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