Abstract
A morphism f: A → B of R-modules is said to be minimal provided that ker f is a superfluous submodule of A. For example, for a right ideal I, the canonical map R → R/I is superfluous if and only if I ⊆ rad R 18.3. A module A is a projective cover (proj. cov.) of B provided that A is projective and there exists a minimal epimorphism A → B. This notion is dual to that of injective hull, and yet, although each R-module has an injective hull, projective covers of modules may fail to exist. For example, as is shown in this chapter, a necessary condition that every right R-module have a projective cover is that R/rad R be semisimple, and rad R be a nil ideal.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).
Björk, J.E.: Rings satisfying a minimum condition on principal ideals. J. reine u. angew. Math. 236, 466–488 (1969).
Björk, J.E.: On subrings of matrix rings over fields. Proc. Cam. Phil. Soc. 68, 275–284 (1970). Brauer and Weiss [64]
Camillo, V. P., Fuller, K.R.: On Loewy length of rings. Pac. J. Math. 53, 347–354 (1974).
Chase, S.U.: Direct products of modules. Trans. Amer. Math. Soc. 97, 457–473 (1960).
Connell, I.: On the group ring. Canad. J. Math. 15, 650–685 (1963).
Courter, R.C.: Finite direct sums of complete matrix rings over perfect completely primary rings. Canad. J. Math. 21, 430–446 (1969).
Cozzens, J. H.: Homological properties of the ring of differential polynomials. Bull. Amer. Math. Soc. 76, 75–79 (1970).
Dickson, S.E., Fuller, K.R.: Commutative QF-1 Artinian rings are QF. Proc. Amer. Math. Soc. 24, 667–670 (1970).
Dlab
Eilenberg, S.: Homological dimension and syzygies. Ann. of Math. 64, 328–336 (1956). ilenberg, S. (see Cartan).
Fuchs, L.: Abelian Groups (Second Edition), Vol.1. Pergamon, New York 1970.
Fuchs, L.: Torsion preradicals and ascending Loewy series of modules. J. reine u. angew. Math. 239, 169–179 (1970).
Fuller, K.R.: On direct representations of quasi-injectives and quasi-projectives. Arch. Math. 20, 495–502 (1969);
Fuller, K.R.: On direct representations of quasi-injectives and quasi-projectives. Arch. Math. 21, 478 (1970).
Fuller, K.R.: Double centralizers of injectives and projectives over Artinian rings. Ill. J. Math. 14, 658–664 (1970).
Golan, J. S.: Characterization of rings using quasiprojective modules, III. Proc. Amer. Math. Soc. 31 (1972).
Gupta, R.N.: Characterization of rings whose classical quotient rings are perfect rings.
Hinohara
Jonah, D.: Rings with minimum condition for principal right ideals have the maximum condition for principal left ideals. Math. Z. 113, 106–112 (1970).
Kasch, F., Mares, E.A.: Eine Kennzeichnung semi-perfekter Moduln. Nagoya Math. J. 27, 525–529 (1966).
Koehler, A.: Quasi-projective covers and direct sums. Proc. Amer. Math. Soc. 24, 655–658 (1970).
Köthe, G.: Die Struktur der Ringe deren Restklassenring nach dem Radical vollständig reduzibel ist. Math. Z. 32, 161–186 (1930).
Krull, W.: Zur Theorie der Allgemeinen Zahlringe. Math. Ann. 99, 51–70 (1928).
Loewy, A.: Über die vollständig reduciblen Gruppen, die zu einer Gruppe linearer homogener Substitutionen gehören. Trans. Amer. Math. Soc. 6, 504–533 (1905).
Loewy, A.: Über Matrizen und Differentialkomplexe, I. Math. Ann. 78, 1–51, 343–368 (1917).
Loewy, A.: Über Matrizen und Differentialkomplexe, II. Math. Ann. 78, 1–51, 343–368 (1917).
Mares, E. A.: Semiperfect modules. Math. Z. 82, 347–360 (1963).
Michler, G.O.: On quasi-local noetherian rings. Proc. Amer. Math. Soc. 20, 222–224 (1969).
Michler, G.O.: Structure of semiperfect hereditary Noetherian rings. J. Algebra 13, 327–344 (1969).
Michler, G.O.: Idempotent ideals in perfect rings. Canad. J. Math. 21, 301–309 (1969).
Michler, G.O.: Asano orders. Proc. Lond. Math. Soc. 19, 421–443 (1969).
Müller, B. J.: On semiperfect rings. Ill. J. Math. 14, 464–467 (1970).
Müller, B.J.: Linear compactness and Morita duality. J. Algebra 16, 60–66 (1970).
Osofsky, B.L.: Loewy length of perfect rings. Proc. Amer. Math. Soc. 28, 352–354 (1971).
Rant
Cailleau, A, Renault, G.: Etude des modules ∑-injective. C. R. Acad. Sci. Paris 270, 1391–1394 (1970).
Rentschler
Rosenberg, A.: Blocks and centres of group algebras. Math. Z. 76, 209–216 (1961).
Sandomierski, F.: Relative injectivity and projectivity. Ph. D. Thesis. Penna, State U., U. Park 1964.
Shores, T. S.: Decompositions of finitely generated modules. Proc. Amer. Math. Soc. 30, 445–450 (1971).
Shores, T.S.: Loewy series of modules. J. reine u. angew. Math. 265, 183–200 (1974).
Swan, R.: Induced representations and projective modules. Ann. of Math. 71, 552–578 (1960).
Swan, R.G.: Algebraic K-Theory. Lecture Notes in Mathematics, vol.76. Springer, Berlin-Heidelberg-New York 1968.
Teply, M.L.: Homological dimension and splitting torsion theories. Pac. J. Math. 34, 193–205 (1970).
Warfield, R.B., Jr.: Rings whose modules have nice decompositions. Math. Z. 125, 187–192 (1972).
Warfield, R.B., Jr.: Exchange rings and decompositions of modules. Math. Ann. 199, 31–36 (1972).
Jans, J., Wu, L.: On quasi-projectives. Ill. J. Math. 11, 439–448 (1967).
Woods
Zöschinger, H.: Moduln, die in jeder Erweiterung ein Komplement haben. Algebra-Berichte-Seminar Kasch und Pareigis. Math. Inst. München 15, 1–22 (1973).
Albrecht, F.: On projective modules over semi-hereditary rings. Proc. Amer. Math. Soc. 12, 638–639 (1861).
Azumaya [75]
Kaplansky, I.: Projective modules. Ann. of Math. 68, 372–377 (1958).
Krull, W.: Theorie und Anwendung der verallgemeinerten Abelschen Gruppen. Sitzungsber. Heidelberger Akad. 7, 1–32 (1926).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1976 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Faith, C. (1976). Projective Covers and Perfect Rings. In: Faith, C. (eds) Algebra II Ring Theory. Grundlehren der mathematischen Wissenschaften, vol 191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65321-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-65321-6_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-65323-0
Online ISBN: 978-3-642-65321-6
eBook Packages: Springer Book Archive