Rings whose modules have maximal or minimal projectivity domain

Küçük Resim Yok

Tarih

2012

Dergi Başlığı

Dergi ISSN

Cilt Başlığı

Yayıncı

Elsevier

Erişim Hakkı

info:eu-repo/semantics/openAccess

Özet

Using the notion of relative projectivity, projective modules may be thought of as being those which are projective relative to all others. In contrast, a module M is said to be projectively poor if it is projective relative only to semisimple modules. We prove that all rings have projectively poor modules. In fact, every ring even has a semisimple projectively poor module. Properties of projectively poor modules are studied and particular emphasis is given to the study of modules over PCI domains; we note that over such domains when all right ideals are principal most modules seem to be either projective or projectively poor. We consider rings over which modules are either projective or projectively poor and call them rings without a p-middle class. We show that a QF ring R with homogeneous right socle and J(R)(2) = 0 has no right p-middle class. As we analyze the structure of rings with no right p-middle class, among other results, we show that any such ring is the ring direct sum of a semisimple artinian ring and a ring K which is either zero or an indecomposable ring such that either (i) K is a semiprimary right SI-ring with J(K) not equal 0, or (ii) K is a semiprimary ring with Soc(K-K) = Z(r)(K) = J(K) not equal 0, or (iii) K is a prime ring with Soc(K-K) = 0, and either f (K) = 0 or (K)J(K) and J(K)(K) are infinitely generated, or (iv) K is a prime right SI-ring with infinitely generated right socle. (C) 2011 Elsevier B.V. All rights reserved.

Açıklama

Anahtar Kelimeler

Kaynak

Journal of Pure and Applied Algebra

WoS Q Değeri

Q3

Scopus Q Değeri

Q1

Cilt

216

Sayı

3

Künye