Automorphism-extendable and endomorphism-extendable modules

A. A. Tuganbaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 4, pp. 175-248.

Résumé

This review paper is concerned with modules in which all automorphisms (endomorphisms) of submodules can be extended to endomorphisms of the entire module. We consider old results and obtain a number of new results. We also consider automorphism-invariant, quasi-injective, and injective modules.

Bibliographie

[1] Tuganbaev A. A., “Stroenie modulei, blizkikh k in'ektivnym”, Sib. matem. zhurn., 18:4 (1977), 890–898 | MR

[2] Tuganbaev A. A., “Kharakterizatsii kolets, ispolzuyuschie maloin'ektivnye i maloproektivnye moduli”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1979, no. 3, 48–51

[3] Tuganbaev A. A., “Koltsa, nad kotorymi vse tsiklicheskie moduli maloin'ektivny”, Tr. seminara im. I. G. Petrovskogo, 6, 1981, 257–262

[4] Tuganbaev A. A., “Tselozamknutye koltsa”, Matem. sb., 115:4 (1981), 544–559 | MR

[5] Tuganbaev A. A., “Maloin'ektivnye moduli”, Matem. zametki, 31:3 (1982), 447–456 | MR

[6] Tuganbaev A. A., “Koltsa s maloin'ektivnymi tsiklicheskimi modulyami”, Abelevy gruppy i moduli, 1986, 151–158

[7] Tuganbaev A. A., “Koltsa s maloin'ektivnymi faktor-koltsami”, Izv. vyssh. uchebn. zaved. Matematika, 1991, no. 1, 80–88

[8] Tuganbaev A. A., “Koltsa, nad kotorymi vse tsiklicheskie moduli vpolne tselozamknuty”, Diskret. matem., 23:3 (2011), 120–137 | DOI

[9] Tuganbaev A. A., “Avtomorfizm-invariantnye moduli”, Fundament. i prikl. matem., 18:4 (2013), 129–135 | MR

[10] Tuganbaev A. A., “Avtomorfizmy podmodulei i ikh prodolzhenie”, Diskret. matem., 25:1 (2013), 144–151 | DOI

[11] Tuganbaev A. A., “Prodolzheniya avtomorfizmov podmodulei”, Fundament. i prikl. matem., 18:3 (2013), 179–198

[12] Tuganbaev A. A., “Kharakteristicheskie podmoduli in'ektivnykh modulei”, Diskret. matem., 25:2 (2013), 85–90 | DOI | MR

[13] Tuganbaev A. A., “Kharakteristicheskie podmoduli in'ektivnykh modulei nad strogo pervichnymi koltsami”, Diskret. matem., 26:3 (2014), 121–126 | DOI

[14] Tuganbaev A. A., “Avtomorfizm-prodolzhaemye moduli”, Diskret. matem., 27:2 (2015), 106–111 | DOI

[15] Feis K., Algebra: koltsa, moduli i kategorii, v. 1, Mir, M., 1979 | MR

[16] Feis K., Algebra: koltsa, moduli i kategorii, v. 2, Mir, M., 1979 | MR

[17] Alahmadi A., Er N., Jain S. K., “Modules which are invariant under monomorphisms of their injective hulls”, J. Aust. Math. Soc., 79:3 (2005), 349–360 | DOI | MR

[18] Anderson F., Fuller K. R., Rings and Categories of Modules, Springer, Berlin, 1973 | MR

[19] Camillo V., “Distributive modules”, J. Algebra, 36:1 (1975), 16–25 | DOI | MR

[20] Dickson S. E, Fuller K. R., “Algebras for which every indecomposable right module is invariant in its injective envelope”, Pacific J. Math., 31:3 (1969), 655–658 | DOI | MR

[21] Dung N. V., Huynh D. V., Smith P. F., Wisbauer R., Extending Modules, Wiley, New York, 1994 | MR

[22] Eisenbud D., Griffith P., “Serial rings”, J. Algebra, 17 (1971), 389–400 | DOI | MR

[23] Er N., “Rings whose modules are direct sums of extending modules”, Proc. Amer. Math. Soc., 137:7 (2009), 2265–2271 | DOI | MR

[24] Er N., Singh S., Srivastava A. K., “Rings and modules which are stable under automorphisms of their injective hulls”, J. Algebra, 379 (2013), 223–229 | DOI | MR

[25] Goodearl K. R., Ring Theory, Marcel Dekker, New York, 1976 | MR

[26] Goodearl K. R., Warfield R. B., An Introduction to Noncommutative Noetherian Rings, Cambridge Univ. Press, Cambridge, 1989 | MR

[27] Gordon R., Robson J. C., “Krull dimension”, Mem. Amer. Math. Soc., 133 (1973), 1–78 | MR

[28] Guil Asensio P. A., Srivastava A. K., “Automorphism-invariant modules satisfy the exchange property”, J. Algebra., 388 (2013), 101–106 | DOI | MR

[29] Handelman D., “Strongly semiprime rings”, Pacific J. Math., 60:1 (1975), 115–122 | DOI | MR

[30] Handelman D., Lawrence J., “Strongly prime rings”, Trans. Amer. Math. Soc., 211 (1975), 209–223 | DOI | MR

[31] Jain S. K., Singh S., “Quasi-injective and pseudo-injective modules”, Can. Math. Bull., 18:3 (1975), 359–366 | DOI | MR

[32] Jain S. K., Srivastava A. K., Tuganbaev A. A., Cyclic Modules and the Structure of Rings, Oxford Univ. Press, Oxford, 2012 | MR

[33] Jeremy L., “Modules et anneaux quasi-continus”, Can. Math. Bull., 17:2 (1974), 217–228 | DOI | MR

[34] Johnson R. E., Wong F. T., “Quasi-injective modules and irreducible rings”, J. London Math. Soc., 36 (1961), 260–268 | DOI | MR

[35] Kutami M., Oshiro K., “Strongly semiprime rings and nonsingular quasi-injective modules”, Osaka J. Math., 17 (1980), 41–50 | MR

[36] Lam T. Y., Exercises in Classical Ring Theory, Springer, New York, 1995 | MR

[37] Lawrence J., “A singular primitive ring”, Trans. Amer. Math. Soc., 45:1 (1974), 59–62 | MR

[38] Lee T. K., Zhou Y., “Modules which are invariant under automorphisms of their injective hulls”, J. Algebra Appl., 6:2 (2013), 1250159 | DOI | MR

[39] Lenagan T. H., “Bounded hereditary Noetherian prime rings”, J. London Math. Soc., 6 (1973), 241–246 | DOI | MR

[40] Mohamed S. H., Müller B. J., Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge, 1990 | MR

[41] Rowen L. H., Ring Theory, v. I, Academic Press, Boston, 1988 | MR

[42] Singh S., “On pseudo-injective modules”, Riv. Mat. Univ. Parma, 9 (1969), 59–65 | MR

[43] Singh S., “Quasi-injective and quasi-projective modules over hereditary Noetherian prime rings”, Can. J. Math., 26:5 (1974), 1173–1185 | DOI | MR

[44] Singh S., “Modules over hereditary Noetherian prime rings”, Can. J. Math., 27:4 (1975), 867–883 | DOI | MR

[45] Singh S., Jain S. K., “On pseudo-injective modules and self-pseudo-injective rings”, J. Math. Sci., 2 (1967), 23–31 | MR

[46] Singh S., Srivastava A. K., “Rings of invariant module type and automorphism-invariant modules”, Ring Theory and Its Applications, Contemp. Math., 609, Amer. Math. Soc., Providence, 2014, 299–311 | DOI | MR

[47] Small L. W., “Semihereditary rings”, Bull. Amer. Math. Soc., 73:5 (1967), 656–658 | DOI | MR

[48] Stenström B., Rings of Quotients, Springer, Berlin, 1975 | MR

[49] Stephenson W., “Modules whose lattice of submodules is distributive”, Proc. London Math. Soc., 28:2 (1974), 291–310 | DOI | MR

[50] Tachikawa H., “QF-3 rings and categories of projective modules”, J. Algebra, 28:3 (1974), 408–413 | DOI | MR

[51] Terlu M. L., “Pseudo-injective modules which are not quasiinjective”, Proc. Amer. Math. Soc., 49:2 (1975), 305–310 | DOI | MR

[52] Tuganbaev A., Semidistributive Modules and Rings, Kluwer Academic, Dordrecht, 1998 | MR

[53] Tuganbaev A. A., “Modules over strongly prime rings”, J. Algebra Appl., 14:5 (2015), 1550076 | DOI | MR

[54] Tuganbaev A. A., “Automorphism-extendable semi-Artinian modules”, J. Algebra Appl., 16:2 (2017), 1750029 | DOI | MR

[55] Tuganbaev A. A., “Automorphism-invariant non-singular rings and modules”, J. Algebra, 485 (2017), 247–253 | DOI | MR

[56] Tuganbaev A. A., Modules over strongly semiprime rings, arXiv: 1701.07117

[57] Tuganbaev A. A., “Injective and automorphism-invariant non-singular modules”, Commun. Algebra (to appear) | MR

[58] Wisbauer R., Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991 | MR