Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của trường đại học vinh năm 2009 tác giả: 12. Ngô Sỹ Tùng, Trần Giang Nam, Ngô Hà Châu Loan, Môđun tựa cấu xạ. | QUASI-MORPHIC MODULES NGQ sy TUNG a TRAN GIANG NAM b NGQ HA CHAU LQAN c Abstract. In this paper we prove that the matrix ring Mn R is left quasi-morphic if and only if the left R-module Rn is quasi-morphic. Then we consider a general problem For a quasi-morphic module M when is end M left quasi-morphic and conversely Using this result we show that a ring R is regular if and only if it is a left quasi morphic left PP ring. 1 INTRODUCTION Nicholson - Campos 5 p. 2630 call a left module M morphic if M Im à ker a for all endomorphism a in end M equivalently if there exists p 2 end M such that Im p ker a and Im a ker p . In this paper we only need the existence of p and 7 such that Im p ker a and Im a ker 7 and we call M quasi-morphic if for every element a in end M a satisfies the above condition. We use the notion to characterize the classes of quasi morphic rings these rings were introduced and studied by Camillo - Nicholson in 2 . More precisely we answer the question For each a ring R when is the ring Mn R left quasi morphic And we obtain that the matrix ring Mn R is left quasi-morphic if and only if Rn is quasi-morphic as a left R-module Theorem . More generally we investigate when M being quasi-morphic implies that end M is left quasi-morphic and conversely Proposition . Furthermore we also investigate when M being quasi-morphic implies that end M is regular Theorem . Then applying this result we obtain that a ring R is regular if and only if it is a left quasi morphic left PP ring Corollary . Throughout this paper every ring R is associative with unity and all modules are unitary. We denote left annihilator of a set X c R by Ir X . Finally all notions used here without any comments can be found in 4 . 1 Nhận bài ngày 08 4 2009. Sửa chữa xong ngày 03 6 2009 2 QUASI MORPHIC MODULES Recall 5 p. 2630 that a left module M is called morphic if M Im a ker a for all endomorphism a in end M equivalently if for each a 2 end M there exists p 2 end M such .
