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A Study of CS and Σ-CS Rings and Modules

Al-Hazmi, Husain S.

Abstract Details

2005, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).

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 Σ-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 López-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 Σ-min-CS if every finitely generated right ideal of R is min CS. Jain, Kanwar and López-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 Σ-CS module is a direct sum of uniform modules and countably Σ-CS modules need not be Σ-CS. A sufficient condition that guarantees a countably Σ-CS module, which is a direct sum of uniform modules, to be Σ-CS has been obtained.

S. Jain (Advisor)
69 p.

Recommended Citations

Citations

  • Al-Hazmi, H. S. (2005). A Study of CS and Σ-CS Rings and Modules [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1121268376

    APA Style (7th edition)

  • Al-Hazmi, Husain. A Study of CS and Σ-CS Rings and Modules. 2005. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1121268376.

    MLA Style (8th edition)

  • Al-Hazmi, Husain. "A Study of CS and Σ-CS Rings and Modules." Doctoral dissertation, Ohio University, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1121268376

    Chicago Manual of Style (17th edition)