On the splitting of the dual Goldie torsion theory

Abstract

Abstract. The splitting of the Goldie (or singular) torsion theory has been extensively studied. Here we determine an appropriate dual Goldie torsion theory, discuss its splitting and answer in the negative a question proposed bÿ Ozcan and Harmancı as to whether the splitting of the dual Goldie torsion theory implies the ring to be quasi-Frobenius.

Coneat Submodules and Coneat-Flat Modules

“…A ring R is small if and only if E = Rad(E) for every injective R-module E (see, [11,Proposition 3.3…”

Covering Coalgebras and Dual Non-singularity

“…Note that C is finite dimensional and C * is isomorphic to the path algebra associated to Γ. Since by [11,13.25] the right maximal ring of quotients of a right artinian right non-singular ring A is isomorphic End(Soc (A A )), we only need to determine the right socle of C * to describe the right maximal ring of quotients of C * . Let A be the path algebra associated to Γ. Denote by Γ sink the set of terminal vertices i ∈ Γ 0 , i.e.…”

“…Lambek's torsion theory is the right concept for a module theoretic setting in which the construction of maximal dense extension of modules are put. Dual Goldie torsion theories have been studied by various authors [17], [9], [13]. As singular modules play the rôle of torsion modules, small modules will play a similar rôle in the dual situation.…”

Lifting Modules

“…References. Bican, Kepka, and Nemec [41]; Jambor, Malcolmson, and Shapiro [181]; Kashu [196]; Lomp [226,228]; Mbuntum and Varadarajan [230]; McMaster [232]; Ohtake [265]; Raggi, Ríos Montes, and Wisbauer [290]; Stenström [323].…”

“…Prove that C ¢ R corresponds to the hereditary torsion class generated by the class of small simple modules ( [291]). [196]; Lomp [226,228]; Mishina and Skornjakov [234]; Ramamurthi [291]; Stenström [323].…”