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The equation XA-BX = I has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where B2 = B, the study of XA - BX = I in a noncommutative ring R leads to several new ways to view and to work with the exchange (or "suitable") elements in R in the sense of Nicholson. For any exchange element A ∊ R, we show that the set of idempotents E ∊ R such that E ∊ RA and I - E ∊ R(I - A) is naturally parametrized by the roots of a certain left-right symmetric "exchange polynomial" associated with A. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood. [ABSTRACT FROM AUTHOR]

In 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring. [ABSTRACT FROM PUBLISHER]

In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units. [ABSTRACT FROM AUTHOR]

Examines the endomorphism rings of the Harada modules. Proof of the Kasch's Theorem using the property of Harada modules; Direct summands of the Harada module; Notations and terminology on the full linear ring.