tailieunhanh - Báo cáo toán học: "New Characterizations and Generalizations of PP Rings"

Bài viết này bao gồm hai phần. Trong phần đầu tiên, nó được chứng minh rằng một R vòng là đúng PP nếu và chỉ nếu tất cả các R mô-đun phải có một bao gồm PI-monic, PI biểu thị các lớp học của tất cả các P-nội xạ phải R-module. Trong phần thứ hai, cho một tập hợp con nonempty X của một R vòng, chúng tôi giới thiệu khái niệm về vòng X-PP thống nhất vòng PP, PS nhẫn và vòng nonsingular. | Vietnam Journal of Mathematics 33 1 2005 97-110 V Í e It ini ai m J o mt r im ai I of MATHEMATICS VAST 2005 New Characterizations and Generalizations of PP Rings Lixin Mao1 2 Nanqing Ding1 and Wenting Tong1 1 Department of Mathematics Nanjing University Nanjing 210093 P. R. China 2 Department of Basic Courses Nanjing Institute of Technology Nanjing 210013 . China Received Febuary 8 2004 Revised December 28 2004 Abstract. This paper consists of two parts. In the first part it is proven that a ring is right if and only if every right -module has a monic -cover where denotes the class of all -injective right -modules. In the second part for a nonempty subset of a ring we introduce the notion of - rings which unifies rings rings and nonsingular rings. Special attention is paid to - rings where is the Jacobson radical of .It is shown that right - rings lie strictly between right rings and right rings. Some new characterizations of von Neumann regular rings and semisimple Artinian rings are also given. 1. Introduction 98 Lixin Mao Nanqing Ding and Wenting Tong . . . . . max. e 2. New Characterizations of PP Rings .cotorsion theory . 1 .preenvelope. F t C. g .envelope. . special C -preenvelope. x. special . . .cyclically presented New Characterizations and Generalizations of Rings 99 .-injective 1 . . . cyclically covered. N . N 0 N 1 N . . . . . . . . . . . . . . . Lemma . Let. . be closed under cokernels of monomorphisms. If then Ext .for any N . . and any integer . Proof. . .. .1. 2 Theorem . The following are equivalent for a ring . is a right. ring Every quotient module of any . - injective right -module is . -injective Every quotient module of any injective right -module has a monic . . -cover. .is closed under cokernels of monomorphisms and every cyclically covered right -module has a monic . . -cover. .is closed under cokernels of monomorphisms every cyclically covered cyclically presented right -module .is closed under .