Rings whose modules have maximal or minimal projectivity domain

Abstract

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 ) ≠ 0 , or (ii) K is a semiprimary ring with S o c ( K K ) = Z r ( K ) = J ( K ) ≠ 0 , or (iii) K is a prime ring with S o c ( K K ) = 0 , and either J ( K ) = 0 or J K ( K ) and J ( K ) K are infinitely generated, or (iv) K is a prime right SI-ring with infinitely generated right socle.