Upper bounds for noetherian dimension of all injective modules with Krull dimension

Hashemi Zadeh Dezfuli, S.J.; Zholanezhad, M.

Upper bounds for noetherian dimension of all injective modules with Krull dimension

Document Type : Research Paper

Authors

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

Abstract

‎In this paper we give an upper bound for Noetherian dimension of all injective modules with Krull dimension on arbitrary rings‎. ‎In particular‎, ‎we also give an upper bound for Noetherian dimension of all Artinian modules on Noetherian duo rings.

Keywords

Main Subjects

16-XX Associative rings and algebras

References

L. Chambless, N-Dimension and N-critical modules, application to artinian modules, Comm. Algebra 16 (1980) 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.

K. R. Goodearl and A.H. Schofield, Non-Artinian essential extensions of simple modules, Proc. Amer. Math. Soc. 97 (1986) 233--236.

K.R. Goodearl and R.B. Warfield, An Introduction to Noncommutative Rings, Cambridge Univ. Press, Cambridge, 1989.

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) 650--662.

J.P. Jans, On co-Noetherian rings, J. Lond. Math. Soc. (2) 1 (1969) 588--590.

N.S. Karamzadeh, O.A.S. Karamzadeh, On Artinian modules over Duo rings, Comm. Algebra 38 (2010) 3521--3531.

O.A.S. Karamzadeh, On maximal right ideals which are direct summands, Bull. Iranian Math. Soc. 10 (1983), no. 2, 40--46.

O.A.S. Karamzadeh and Sh. Rahimpour, On λ-finitely embedded modules, Algebra. Colloq. 12 (2005) 281--292.

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.

D. Kirby, Dimension and length for Artinian modules, Q. J. Math. 41 (1990) 419--429.

T.Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math. Springer Verlag, New York, 1999.

J. Lambek and G. Michler, The torsion theory at a prime ideal of a right Noetherian ring, J. Algebra 25 (1973) 364--389.

B. Lemonnier, Dimension de Krull et codeviation, application au theorem d'Eakin, Comm. Algebra 6 (1978) 1647--1665.

E. Matlis, Modules with descending chain condition, Trans. Amer. Math. Soc. 97 (1960) 495--508.

J.C. McConnel and J.C. Robson, Noncommutative Noetherian Rings, Revised edition, Grad. Stud. Math. 30, Amer. Math. Soc. Providence, RI, 2000.

M.L. Reyes, A one-sided prime ideal principle for noncommutative rings, J. Algebra Appl. 9 (2010) 877--919.

R.Y. Sharp, A method for the study of Artinian modules with an application to asymptotic behavior, in: Commutative Algebra (Berkeley, CA, 1987), pp. 443--465, Math. Sci. Res. Inst. Publ. 15, Springer Verlag, New York, 1989.

R.Y. Sharp, Steps in Commutative Algebra, Cambridge Univ. Press, 1990.

P. Vamos, The dual of the notion of finitely generated, J. Lond. Math. Soc. 43 (1968) 643--646.