A note on essential extensions of semi-simple modules

Er N. F.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.7, no.2, pp.225-230, 2008 (SCI-Expanded) identifier identifier


In a series of recent papers, Beidar, Jain and Srivastava studied the question as to when a ring R with the property that essential extensions of semi-simple right R-modules are direct sums of quasi-injectives is right Noetherian. Beidar and Jain proved that it is, when R is commutative or right q. f. d. In this note we extend their results proving the following: A ring R with this property is right Noetherian iff for some n is an element of N, R/soc(n)(R-R) has ascending chain condition on essential non-two-sided right ideals ( in particular, when R/socn( RR) is right q.f.d. or commutative). Also shown is the following: A ring is a right Noetherian right V-ring iff modules with essential socle are quasi-continuous/quasi-injective.