Applications of epi-retractable modules

Document Type : Research Paper


1 Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India.

2 Department of Mathematics, Jaypee Institute of Information Technology, (Deemed University) A-10, Sector-62, Noida-201307 (UP), India


An R-module M is called epi-retractable if every submodule of MR is a homomorphic image of M. It is shown that if R is a right perfect ring, then every projective slightly compressible module MR is epi-retractable. If R is a Noetherian ring, then every epi-retractable right R-module has direct sum of uniform submodules. If endomorphism ring of a module MR is von-Neumann regular, then M is semi-simple if and only if M is epi-retractable. If R is a quasi Frobenius ring, then R is a right hereditary ring if and only if every injective right R-module is semi-simple. A ring R is semi-simple if and only if R is right hereditary and every epiretractable right R-module is projective. Moreover, a ring R is semi-simple if and only if R is a pri and von-Neumann regular.


  • Receive Date: 17 September 2010
  • Revise Date: 19 February 2011
  • Accept Date: 19 February 2011
  • First Publish Date: 15 July 2012