Modules with Artinian prime factors

Armendariz, Efraim P.

doi:10.1090/S0002-9939-1980-0553364-X

Modules with Artinian prime factors
HTML articles powered by AMS MathViewer

by Efraim P. Armendariz PDF

Proc. Amer. Math. Soc. 78 (1980), 311-314 Request permission

Abstract:

An R-module M has Artinian prime factors if $M/PM$ is an Artinian module for each prime ideal P of R. For commutative rings R it is shown that Noetherian modules with Artinian prime factors are Artinian. If R is either commutative or a von Neumann regular V-ring then the endomorphism ring of a module with Artinian prime factors is a strongly $\pi$-regular ring.

References

E. P. Armendariz and Joe W. Fisher, Regular $P.I.$-rings, Proc. Amer. Math. Soc. 39 (1973), 247–251. MR 313305, DOI 10.1090/S0002-9939-1973-0313305-3
Efraim P. Armendariz, Joe W. Fisher, and Robert L. Snider, On injective and surjective endomorphisms of finitely generated modules, Comm. Algebra 6 (1978), no. 7, 659–672. MR 469974, DOI 10.1080/00927877808822263
William D. Blair, Right Noetherian rings integral over their centers, J. Algebra 27 (1973), 187–198. MR 325679, DOI 10.1016/0021-8693(73)90173-7
John Cozzens and Carl Faith, Simple Noetherian rings, Cambridge Tracts in Mathematics, No. 69, Cambridge University Press, Cambridge-New York-Melbourne, 1975. MR 0396660

On strongly

regular rings

285

Joe W. Fisher, Nil subrings of endomorphism rings of modules, Proc. Amer. Math. Soc. 34 (1972), 75–78. MR 292878, DOI 10.1090/S0002-9939-1972-0292878-2
K. R. Goodearl, Artinian and Noetherian modules over regular rings, Comm. Algebra 8 (1980), no. 5, 477–504. MR 561542, DOI 10.1080/00927878008822470
Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, 190 Hope Street, Providence, R.I., 1956. MR 0081264
Robert W. Miller and Darrell R. Turnidge, Some examples from infinite matrix rings, Proc. Amer. Math. Soc. 38 (1973), 65–67. MR 310001, DOI 10.1090/S0002-9939-1973-0310001-3
Morris Orzech, Onto endomorphisms are isomorphisms, Amer. Math. Monthly 78 (1971), 357–362. MR 280475, DOI 10.2307/2316897
Hans Heiner Storrer, Epimorphismen von kommutativen Ringen, Comment. Math. Helv. 43 (1968), 378–401 (German). MR 242810, DOI 10.1007/BF02564404
Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505–512. MR 238839, DOI 10.1090/S0002-9947-1969-0238839-5