Koszul Objects And Weak Regular Ring

Firstly, the minimal Horse-shoe Lemma and the generalized extension closure of Koszul objects are investigated, in this paper. Mainly, we have the following statements:1. We give some sufficient conditions for the minimal Horse-shoe Lemma holding. Moreover, some applications of the minimal Horse-shoe Lemma are given.2. The conception of generalization extension closure which is a wider conception than the classical extension closure is introduced. On this basis, we consider the generalization extension closure of the categories K(A), QK(A), HK(A) and NBGr(A).Secondly, we introduce the concept of Koszul-type algebra, which is a 2p homogeneous algebra and neither a Koszul algebra nor a h.-Koszul algebra, where p ≥ 1 is an integer. Although it is different from Koszul and high Koszul algebras, it does have a lot of perfect homological properties similar to Koszul and high Koszul algebras. Further more, the generalized extension closure of Koszul-type modules is discussed and the one point extension of Koszul-type algebras is investigated as well.Thirdly, we introduce the notion of f-module (algebra) which is the generalizations of Koszul module and high Koszul module (we usually call them Koszul objects); The extension closure of f-modules and the one point extension of f-algebras are discussed.Finally, as a generalization of the group-graded regular ring, the notion of group-graded weakly regular ring is introduced and we discuss it in a new way. In particular, an equivalent condition for the semigroup rings to be weakly regular(T-weakly regular) is given.