On the Noetherian dimension of Artinian modules with homogeneous uniserial dimension

Document Type : Research Paper


Department of Mathematics‎, ‎Shahid Chamran University of Ahvaz‎, ‎Ahvaz‎, ‎Iran.


 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $\Bbb Z_{p^{\infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dimension is less than or equal‎ ‎to $1$‎. ‎In particular‎, ‎if $A$ is a quotient finite dimensional‎ ‎$R$-module with homogeneous uniserial dimension‎, ‎where $R$ is a‎ ‎locally Noetherian (or‎, ‎a Noetherian duo) ring‎, ‎then $n$-dim ‎$A\leq‎ ‎1$‎. ‎We also show that the Krull dimension of Noetherian modules is‎ ‎bounded by the uniserial dimension of these modules‎. ‎Moreover‎, ‎we introduce the concept of qu-uniserial modules and by using this‎ ‎concept‎, ‎we observe that if $A$ is an Artinian $R$-module‎, ‎such that‎ ‎any of its submodules is qu-uniserial‎, ‎where $R$ is a right duo (or‎, ‎a right Noetherian) ring‎, ‎then $n$-dim $‎A\leq 1$.


Main Subjects

T. Albu and S. Rizvi, Chain condition on quotient finite dimensional modules, Comm. Algebra 29 (2001), no. 5, 1909--1928.
T. Albu and P.F. Smith, Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki theorem (I), Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 87--101.
T. Albu and P.F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math. 29 (1999), no. 4, 1153--1165.
T. Albu and L. Teply, Generalized deviation of posets and modular lattices, Discrete Math. 214 (2000), no. 1-3, 1--19.
T. Albu and P. Vamos, Global Krull dimension and global dual Krull dimension of valuation rings, in: Abelian Groups, Module Theory, and Topology (Padua, 1997), pp. 37--54, Lecture Notes in Pure and Appl. Math. 201, Dekker, New York, 1998.
F.W. Anderson and K.R. Fuller, Rings and categories of modules, Grad. Texts Math., 13 , Springer, Berlin, 1992.
L. Chambless, N-Dimension and N-critical modules. Application to Artinian modules, Comm. Algebra. 8 (1980), no. 16, 1561--1592.
M. Davoudian, A. Halali and N. Shirali, On α-almost Artinian modules, Open Math. 14 (2016) 404--413.
M. Davoudian and O.A.S. Karamzadeh, Artinian serial modules over commutative (or, left Noetherian) rings are at most one step away from being Noetherian, Comm. Algebra 44 (2016), no. 9, 3907--3917.
M. Davoudian, O.A.S. Karamzadeh and N. Shirali, On α-short modules, Math. Scand. 114 (2014), no. 1, 26--37.
A. Facchini, Loewy and Artinian modules over Commutative rings, Ann. Mat. Pura Appl. (4) 128 (1980), 359--374.
R. Gordon and J.C. Robson, Krull dimension, Mem. Amer. Math. Soc. 133, Amer. Math. Soc. Providence, RI, 1973.
J. Hashemi, O.A.S. Karamzadeh and N. Shirali, Rings over which the Krull dimension and the Noetherian dimension of all modules coincide, Comm. Algebra 37 (2009), no. 2, 650--662.
Y. Hirano and I. Mogami, Modules whose proper submodules are non-hopf kernels, Comm. Algebra 15 (1987), no. 8, 1549--1567.
O.A.S. Karamzadeh, Noetherian-dimension, PhD. Thesis, Exeter, 1974.
O.A.S. Karamzadeh and M. Motamedi, On α-DICC modules, Comm. Algebra 22 (1994), no. 6, 1933--1944.
O.A.S. Karamzadeh and M. Motamedi, a-Noetherian and Artinian modules, Comm. Algebra 23 (1995), no. 10, 3685--3703.
O.A.S. Karamzadeh and A.R. Sajedinejad, Atomic modules, Comm. Algebra 29 (2001), no. 7, 2757--2773.
O.A.S. Karamzadeh and A.R. Sajedi Nejad, On the Loewy length and the Noetherian dimension of Artinian modules, Comm. Algebra. 30 (2002), no. 3, 1077--1084.
O.A.S. Karamzadeh and N. Shirali, On the countability of Noetherian dimension of modules, Comm. Algebra 32 (2004), no. 10, 4073--4083.
D. Kirby, Dimension and length for Artinian modules, Quart. J. Math. Oxford Ser. (2) 41 (1990), no. 164, 419--429.
G. Krause, On the Krull-dimension of left Noetherian rings, J. Algebra 23 (1972) 88-99.
B. Lemonnier, Deviation des ensembles et groupes abeliens totalement ordonnes, Bull. Sci. Math. (2) 96 (1972) 289--303.
J.C. McConell and J.C. Robson, Noncommutative Noetherian Rings, With the cooperation of L.W. Small, Wiley Ser. Pure Appl. Math. Wiley, Chichester, New York, 1987.
Z. Nazemian, A. Ghorbani and M. Behboodi, Uniserial dimension of modules, J. Algebra 399 (2014) 894--903.
R.N. Roberts, Krull-dimension for Artinian modules over quasi local commutative Rings, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 103, 269--273.
P. Vamos, The dual of the notion of finitely generated, J. Lond. Math. Soc. 43 (1968) 643--646.