A Study On Semi-projective Covers, Semi-projective Modules and Formal Triangular Matrix Rings

We show that a ring R is right perfect if and only if every right R-module has a semi-projective cover. We characterize (semi)hereditary and semisimple rings via semi- projective modules. Finally we investigate the relative projectivity of modules over a formal triangular matrix ring T = " A 0 M B # . We also prove that if a right T -module (X Y )T is lifting, then (X=Y M)A andYB are lifting.

[1]  M. R. Vedadi,et al.  PURE PROJECTIVITY AND PURE INJECTIVITY OVER FORMAL TRIANGULAR MATRIX RINGS , 2012 .

[2]  D. K. Tütüncü,et al.  Direct sums of semi-projective modules , 2012 .

[3]  A. Tuganbaev,et al.  Modules over formal matrix rings , 2010 .

[4]  P. Krylov.  Injective modules over formal matrix rings , 2010 .

[5]  P. Krylov,et al.  Projective and hereditary modules over rings of generalized matrices , 2009 .

[6]  A. Haghany.  Injectivity conditions over a formal triangular matrix ring , 2002 .

[7]  K. Varadarajan,et al.  Study of modules over formal triangular matrix rings , 2000 .

[8]  D. Keskin.  On lifting modules , 2000 .

[9]  Robert Wisbauer,et al.  Modules and algebras - bimodule structure on group actions on algebras , 1996, Pitman monographs and surveys in pure and applied mathematics.

[10]  W. Xue.  CHARACTERIZATIONS OF SEMIPERFECT AND PERFECT RINGS ( , 1996 .

[11]  R. Wisbauer,et al.  Modules with every subgenerated module lifting , 1995 .

[12]  W. Xue.  Characterization of rings using direct-projective modules and direct-injective modules , 1993 .

[13]  Robert Wisbauer,et al.  Foundations of module and ring theory , 1991 .

[14]  T. Faticoni.  On quasiprojective covers , 1983 .

[15]  E. Green.  On the representation theory of rings in matrix form. , 1982 .

[16]  K. Fuller,et al.  On quasi-projective modules via relative projectivity , 1970 .

[17]  A. Koehler.  QUASI-PROJECTIVE COVERS AND DIRECT SUMS , 1970 .

[18]  J. Golan.  Characterization of rings using quasiprojective modules , 1970 .

[19]  J. Jans,et al.  On quasi projectives , 1967 .

[20]  S. Kaye.  Ring Theoretic Properties of Matrix Rings , 1967, Canadian Mathematical Bulletin.

[21]  H. Bass.  Finitistic dimension and a homological generalization of semi-primary rings , 1960 .