On M-injective and M-projective Modules

A left R-module M is called max-injective (or m-injective for short) if for any maximal left ideal I,any homomorphism f : I → M can be extended to g : R → M, if and only if Ext1R(R/I, M) = 0for any maximal left ideal I. A left R-module M is called max-projective (or m-projective for short)if Ext1R(M, N) = 0 for any max-injective left R-module N. We prove that every left R-module has aspecial m-projective precover and a special m-injective preenvelope. We characterize C-rings, SF ringsand max-hereditary rings using m-projective and m-injective modules.

PDF

___

[1] Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting modules. Frontiers in Mathematics, Birkhaauser Verlag, Basel (2006).
[2] Enochs, E.E., Jenda, O.M.G.: Relative homological algebra. Berlin: Walter de Gruyter (2000).
[3] Enochs, E.E., Jenda, O.M.G., Lopez-Ramos, J.A.: The existence of Gorenstein flat covers. Math. Scand. 94(1), 46-62 (2004).
[4] Eklof, P.C., Trlifaj, J.: How to make Ext vanish, Bull. London Math. Soc. 33(12), 41-51 (2001).
[5] Garcia Rozas, J.R., Torrecillas, B.: Relative injective covers, Comm. Algebra, 22(8), 2925-2940 (1994).
[6] Megibben, C.: Absolutely pure modules. Proc. Amer. Math. Soc. 18, 155-158 (1967).
[7] Moradzadeh-Dehkordi, A., Shojaee, S.H.: Rings in which every ideal is pure-projective or FP-projective. J. Algebra, 478, 419-436 (2017).
[8] Ramamurthi, V.S.: On the injectivity and flatness of certain cyclic modules. Proc. Amer. Math. Soc. 48, 21-25 (1975).
[9] Smith, P.F.: Injective modules and prime ideals. Comm. Algebra, 9(9), 989-999 (1981).
[10] Wang, M.Y., Zhao, G.: On maximal injectivity. Acta Math. Sin. 21(6), 1451-1458 (2005).
[11] Xiang, Y.: Max-injective, max-flat modules and max-coherent rings. Bull. Korean Math. Soc. 47(3), 611-622 (2010).

Mathematical Sciences and Applications E-Notes-Cover

ISSN: 2147-6268
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 2013
Yayıncı: -

Arşiv