The length of Artinian modules with countable Noetherian dimension

Document Type : Research Paper


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


‎It is shown that‎ ‎if $M$ is an Artinian module over a ring‎ ‎$R$‎, ‎then $M$ has Noetherian dimension $\alpha $‎, ‎where $\alpha $ is a countable ordinal number‎, ‎if and only if $\omega ^{\alpha }+2\leq \it{l}(M)\leq \omega ^{\alpha‎ +1}$, ‎where $ \it{l}(M)$ is‎ ‎the length of $M$‎, ‎$i.e.,$ the least ordinal number such that the interval $[0‎, ‎\ \it{l}(M))$ cannot be embedded in the lattice of all submodules of $M$.


Main Subjects

T. Albu and S. Rizvi, Chain conditions on quotient finite dimensional modules, Comm. Algebra 29 (2001), no. 5, 1909--1928.
T. Albu and P.F. Smith, Dual Krull dimension and duality, Rocky Mountain J. Math. 29 (1999) 1153--1164.
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.
H. Bass, Descending chains and the Krull ordinal of commutative rings. J. Pure Appl. Algebra 1 (1971) 347--360.
L. Chambless, N-dimension and N-critical modules, application to Artinian modules, Comm. Algebra 8 (1980), no. 16, 1561--1592.
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.
K.R. Goodearl and B. Zimmermann-Huisgen, Lengths of submodule chain versus Krull dimension in Non-Noetherian modules, Math. Z. 191 (1986) 519--527.
R. Gordon, Gabriel and Krull dimension, in: Ring theory (Proc. Conf., Univ. Oklahoma, Norman, Okla. 1973), pp. 241--295, Lecture Notes in Pure and Appl. Math. 7, Dekker, New York, 1974.
R. Gordon and J.C. Robson, Krull dimension, Mem. Amer. Math. Soc. 133 (1973) 78 pages.
J. Hashemi, O.A.S. Karamzadeh and N. Shirali, Rings over which the Krull dimension and Noetherian dimension of all modules coincide, Comm. Algebra 37 (2009) 650--662.
O.A.S. Karamzadeh, Noetherian-Dimension. PhD Thesis, Exeter, 1974.
O.A.S. Karamzadeh, When are Artinian modules countable generated?, Bull. Iran. Math. Soc. 9 (1982) 171--176.
O.A.S. Karamzadeh and M. Motamedi, On α-DICC modules, Comm. Algebra 22 (1994) 1933--1944.
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. Sajedinejad, On the Loewy length and the Noetherian dimension of Artinian modules, Comm. Algebra 30 (2002) 1077--1084.
O.A.S. Karamzadeh and N. Shirali, On the countablity of Noetherian dimension of modules, Comm. Algebra 32 (2004) 4073--4083.
D. Kirby, Dimension and length for Artinian modules, Q. J. Math. (2) 41 (1990) 419--429.
G. Krause, Descending chains of submodules and the Krull dimension of Noetherian modules, J. Pure Appl. Algebra 3 (1973) 385--397.
B. Lemonnier, Deviation des ensembless et groupes totalement ordonnes, Bull. Sci. Math. 96 (1972) 289--303.
B. Lemonnier, Dimension de Krull et codeviation, application au theorem d'Eakin, Comm. Algebra 6 (1978) 1647--1665.
J.C. McConell and J.C. Robson, Noncommutative Noetherian rings, Wiley-Interscience, New York, 1987.
R.N. Roberts, Krull dimension for Artinian modules over quasi local commutative rings, Q. J. Math. 26 (1975) 269--273.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991.