A study of CS and Sigma-CS rings and modules

A right R-module M is called CS if every submodule of M is essential in a direct summand of M. In this dissertation, we study certain classes of CS and Sigma-CS rings and modules. A ring R is called right (left) max-min CS if every maximal closed right (left) ideal with nonzero left (right) annihilator and every minimal closed right (left) ideal of R is a direct summand of R. Among other results, it is shown that if R is a nondomain prime ring, then R is right nonsingular, right max-min CS with a uniform right ideal if and only if R is a left nonsingular, left max-min CS with a uniform left ideal. This result gives, in particular, Huynh, Jain and Lopez-Permouth Theorem for prime rings of finite uniform dimension. Also we show that a nondomain right nonsingular prime ring with a uniform right ideal is right finitely Sigma-min- CS if every finitely generated right ideal of R is min CS. Jain, Kanwar and Lopez-Permouth characterized right nonsingular semiperfect right CS rings. We obtain the structure of right nonsingular semiperfect right min CS rings with a uniform right ideal. It is shown that such rings are direct sums of indecomposable right CS rings and a ring with no uniform right ideal. As a consequence, we show that an indecomposable right nonsingular semiperfect ring is right CS if and only if it is min CS with a uniform right ideal. We generalize this result to endomorphism rings of nonsingular semiperfect progenerator min CS modules with a uniform submodule. It is known that every Sigma-CS module is a direct sum of uniform modules and countably Sigma-CS modules need not be Sigma-CS. A sufficient condition that guarantees a countably Sigma-CS module, which is a direct sum of uniform modules, to be Sigma-CS has been obtained.