Strongly N-Ding Modules And The Related Properties

Since Mahdou and Tamekkante introduced the notions of strongly n-Gorenstein projective, injective and flat modules and investigated their properties, inspired by this, we define strongly n-Ding projective, injective and flat modules, and study some basic properties in this paper. This paper is divided into four parts:In Section1, we introduce the background. In Section2, we collect some basic notions and facts about Ding modules and strongly Ding modules. In Section3, we introduce the notions of strongly n-Ding modules, and give the main results of this paper. We prove that every module with projective dimension at most n is strongly n-Ding projective; every module with injective dimension at most n is strongly n-Ding injective. Moreover, the class of strongly n-Ding projective modules is closed under direct sums; the class of strongly n-Ding injective modules is closed under direct products. In particular, we prove that the Ding projective dimension of a module M is at most n if and only if M is a direct summand of some strongly n-Ding projective module, the Ding injective dimension of a module M is at most n if and only if M is a direct summand of some strongly n-Ding injective module, which is a generalization to that of strongly Ding modules. Meanwhile, we discuss some relations between strongly n-Ding projective, injective and flat modules, and also show the connections between strongly n-Ding modules and strongly n-Gorenstein modules. In Section4, we consider some results of strongly n-Ding modules over commutative rings.