When is the ring of real measurable functions a hereditary ring?

Document Type : Research Paper

Authors

Department of Mathematical Sciences‎, ‎Shahrekord‎ ‎University‎, ‎P.O. Box 115‎, ‎Shahrekord‎, ‎Iran.

Abstract

‎Let $M(X‎, ‎\mathcal{A}‎, ‎\mu)$ be the ring of real-valued measurable functions‎ ‎on a measure space $(X‎, ‎\mathcal{A}‎, ‎\mu)$‎. ‎In this paper‎, ‎we characterize the maximal ideals in the rings of real measurable functions‎ ‎and as a consequence‎, ‎we determine when $M(X‎, ‎\mathcal{A}‎, ‎\mu)$ is a hereditary ring.

Keywords

Main Subjects

References

H. Azadi, M. Henriksen and E. Momtahen, Some properties of algebras of real-valued mesurable functions, Acta Math. Hungar. 124 (2009), no. 1-2, 15--23.
F. Azarpanah and O.A.S. Karamzadeh, Algebraic characterization of some disconnected spaces, Ital. J. Pure Appl. Math. 12 (2002) 158--168.
F. Azarpanah, O.A.S. Karamzadeh and S. Rahmati, C(X) vs. C(X) modulo its socle, Colloq. Math. 111 (2008), no. 2, 315--336.
G.M. Bergman, Hereditary commutative rings and center of hereditary rings, Proc. Lond. Math. Soc. 23 (1971) 214--236.
J.G. Brookshear, Projective ideals in rings of continuous functions, Pacific J. Math. 71 (1977) 313--333.
J.G. Brookshear, On projective prime ideals in C(X), Proc. Amer. Math. Soc. 69 (1978) 203--204.
R. Engelking, General Topology, Heldermann-Verlag, 1989.
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1989.
K.R. Goodearl, Von Neumann Regular Rings, Krieger Publ. Co. 1991.
A.W. Hager, Algebras of measurable functions, Duke Math. J. 38 (1971) 21--27.
P.R. Halmous, Measure Theory, Springer-Verlag, New York, 1973.
O.A.S. Karamzadeh, Projective maximal right idals of self-injective rings, Proc. Amer. Math. Soc. 48 (1975), no. 2, 286--288.
O.A.S. Karamzadeh, On maximal right ideals which are dirct summands, Bull. Iranian Math. Soc. 10 (1983), no. 2, 40--46.
O.A.S. Karamzadeh, On a qustion of Matlis, Comm. Algebra 25 (1997), no. 9, 2717--2726.
E. Momtahen, Essential ideals in rings of measurable functions, Comm. Algebra 38 (2010), no. 12, 4739--4746.
J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
W. Rudin, Real and Complex Analysis, Springer-Verlag, New York, 1985.
W.V. Vasconcelos, Finiteness in projective ideals, J. Algebra 25 (1973) 269--278.

History

• Receive Date: 01 June 2015
• Revise Date: 19 November 2016
• Accept Date: 23 November 2016
• First Publish Date: 30 November 2017