ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 16 (2021), 237 -- 249
This work is licensed under a Creative Commons Attribution 4.0 International License.
A NOTE ON PROJECTIVE MODULES
Hossein Faridian
Abstract. This expository note delves into the theory of projective modules parallel to the one developed for injective modules by Matlis. Given a perfect ring R, we present a characterization of indecomposable projective R-modules and describe a one-to-one correspondence between the projective indecomposable R-modules and the simple R-modules.
2020 Mathematics Subject Classification: 16L30; 16D40; 13C05.
Keywords: indecomposable module; perfect ring; projective module; sum-irreducible module.
References
F.W. Anderson and K.R. Fuller, Rings and categories of modules, Second edition, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1992. MR1245487.
M. Auslander, I. Reiten and S.O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1995. MR1314422.
E.E. Enochs and O.M.G. Jenda, Relative homological algebra, Volume 1, Second edition, De Gruyter Exposition in Mathematics, 30, Walter de Gruyter GmbH & Co. KG, Berlin, 2011.
T. Leinster, The bijection between projective indecomposable and simple modules, Bull. Belg. Math. Soc. Simon Stevin, 22(5), (2015), 725-735. MR3435078. Zbl 1334.16002.
E. Matlis, Injective modules over Noetherian rings, Pacific J. Math., 8, (1958), 511-528. MR0099360. Zbl 0084.26601.
U. Shukla, On the projective cover of a module and related results, Pacific J. Math. 12 (1962), 709-717. MR0146235. Zbl 0109.02701.
J. Xu, Flat covers of modules, Lecture Notes in Mathematics 1634, Springer-Verlag, Berlin, 1996. MR1438789.
Hossein Faridian
School of Mathematical and Statistical Sciences, Clemson University, SC 29634, USA.
email: hfaridi@g.clemson.edu