Semiregular, weakly regular, and π-regular rings

Abstract

Publisher Summary This chapter discusses the semi-regular, weakly regular and π-regular rings. All rings are associative and have nonzero identity elements. Every strongly π-regular ring is π-regular. The factor ring of the integers with respect to the ideal generated by the integer four is a strongly re-regular ring which is not a regular ring. For a ring A, the Jacobson radical of A is denoted by J (A) which is equal to the intersection of all maximal right ideals of A by definition. A ring A is said to be semi-regular if A / J (A) is a regular ring, and all idempotents of A / J (A) are images of idempotents of A for the natural epimorphism A → A / J (A). The classes of semi-regular and π-regular rings are quite large. Every regular ring is a semi-regular π-regular ring. All right or left Artinian rings are semi-regular π-regular rings, and the endomorphism ring of any injective module is a semi-regular ring.