Rings Characterized by Properties of Direct Sums of Modules and on Rings Generated by Units

The study of rings over which the direct sums of modules have certain properties is a well recognized topic for research in Ring Theory and Homological Algebra. In this dissertation, the class of rings over which every essential extension of a direct sum of simple right modules is a direct sum of quasi-injective right modules is studied. It is shown that under this condition on a ring R, (i) R must be directly finite, (ii) R has bounded index of nilpotence if R is also right non-singular, and (iii) R is right noetherian when R is semi-regular in the sense that R/J(R) is a von-Neumann regular ring.

This dissertation initiates the study of rings having the property that each right ideal is a finite direct sum of quasi-injective right ideals. These rings have been named as right Nakayama-Fuller rings (in short, NF-rings). Prime right self-injective right NF- rings are shown to be simple artinian. Right artinian right non-singular right NF- rings are shown to be upper triangular block matrix rings over rings which are either zero rings or division rings. Examples are provided to show that the NF- rings are not left-right symmetric nor they are Morita invariant.

Carl Faith, Cailleau, Megibben and others have studied Σ-injective module M in the sense that every direct sum of copies of M is injective. In this dissertation, a new characterization for an injective module to be Σ-injective has been provided. This leads to a new characterization of right noetherian rings which generalizes results of Cartan-Eilenberg, Bass and Beidar et al.

Zelinsky proved that every element in the ring of linear transformations of a vector space V over a division ring D is a sum of two units unless dim V = 1 and D = ℤ₂. Zelinsky's result has been extended to include all the previous known results by proving that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to ℤ₂, thus answering a long-standing question on a characterization of right self-injective ring generated by units.