SF-Rings And Regular Rings

The main purpose of this paper is to study the regularity of SF-rings and we prove that SF-rings that satisfy certain additional conditions are regular rings. Throughout this paper have four chapters. In the first chapter, we study the strong regularity of SF-rings whose every maximal one-sided ideal is generalized weak ideal and prove that R is strongly regular ring if and only if R is a left SF-ring whose every maximal left(right) ideal is a GW-ideal. In the second chapter, we study the regularity of SF-rings whose every maximal essential one-sided ideal is generalized weak ideal. It is proved that if R is a left(right) SF-ring whose every maximal essential left(right) ideal is a GW-ideal, then R is a Von Neumann regular ring. In the third chapter, we study the strong regularity of SF-rings whose every complement one-sided ideal is weak ideal. It is proved that if R is a left(right) SF-ring whose every complement left (right) ideal is a W-ideal, then R is a strongly regular ring. In the fourth chapter, we study the strong regularity of left SF-rings whose every nonzero one-sided ideal contains nonzero weak ideal and every special annihilator is weak ideal, and we prove the following results: l)Strong left(right) weakly bounded left SF-rings are strongly regular rings. 2)Let R is a semiprime ring, if R is a left SF-ring whose every special right annihilator is W-ideal, then R is a strongly regular ring. 3)If R is a left SF-ring whose every special left annihilator is W-ideal, then R is a strongly regular ring.