| For many years,the morphic property of rings and modules have been the research object of algebraic scholars.With the continuous deepening of research,many scholars have began to generalize the morphic of rings and modules from different directions.The main content of this article is to study the subclasses of morphic modules,namely,exact-morphic modules,and to generalize morphic of rings and modules from another perspective,thereby obtaining some meaningful results.In Chapter 1,we introduce the research background and sources of this topic,briefly summarize the main conclusions of this article,as well as the basic definitions and symbols used in this article.In Chapter 2,we give the definition and equivalent characterization of exact-morphic modules,then we define strong exact-morphic modules,obtain some properties of strong exact-morphic modules,and give a condition for the equivalence of morphic modules and exact-morphic modules.Finally,we obtain the characterization that a finitely generated abelian group is exact-morphic.In Chapter 3,we define morphic-idempotent rings and morphic-idempotent modules,and use examples to illustrate that they are true generalizations of morphic rings and morphic modules,respectively.The first section mainly studies some properties of left morphicidempotent rings,characterize left morphic-idempotent elements in rings from the perspective of trivial extensions,and study left morphic-idempotent properties of formal triangular matrix rings.The second section mainly studies some properties of left morphic-idempotent modules,we give a condition for the left morphic-idempotent modules to maintains the direct sum.Finally,we discuss the relationship between left morphic-idempotent modules and its endomorphism rings. |
