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Discrete Koszul algebra, another extension of Koszul algebras, is introduced in this paper. The Yoneda algebra of a discrete Koszul algebra is investigated in detail. As an application, we give an answer to a question proposed by Green and Marcos (Commun Algebra 33:1753–1764, 2005). In particular, the relationship between discrete Koszul algebras and Koszul algebras is established. Further, we construct new discrete Koszul algebras from the given ones in terms of one-point extension.

We introduce the concept of ideal–comparability condition for regular rings. Let I be an ideal of a regular ring R. If R satisfies the I–comparability condition, then R is one–sided unit–regular if and only if so is R / I. Also, we show that a regular ring R satisfies the general comparability if and only if the following hold: (1) R / I satisfies the general comparability; (2) R satisfies the general I–comparability condition; (3) The natural map B(R) → B(R / I) is surjective.

In this paper, we prove that if R is a Min-E ring, then the following statements are equivalent: (1) Every left primitive factor ring of R is left artinian; (2) R is a π-regular ring; (3) R is an Exchange ring; (4) R is a Clean ring. As an application, we obtain that if R is Min-E ring, every left primitive factor ring of R is artinian and any direct product of them is a Min-E ring, then every non-zero homomorphic image of R contains only a finite number of prime ideals. When R is Min-E ring and has no infinite set of orthogonal idempotents, a left R-module M is Min-E if and only if M is Max-E. Also, we show, if R is a strongly clean Min-E ring, then R is directly finite and has stable range 1.

In this paper, we shall discuss the conditions for a right SC right CS ring to be a QF ring. In particular, we prove that if R is a right SI right CS ring satisfying the reflexive orthogonal condition (*) and if every CS right R-module is Σ-CS, then R is a QF ring.