Authors:G. F. Birkenmeier, Karabacak F. and Tercan A.
A right R-module M has right SIP (SSP) if the intersection (sum) of two direct summands of M is also a direct summand. It is shown that the right SIP (SSP) is not a Morita invariant property and that a nonsingular
C11+-module does not necessarily have SIP. In contrast, it is shown that the direct sum of two copies of a right Ore domain has
SIP as a right module over itself.
Authors:G.F. Birkenmeier, G.L. Booth and N.J. Groenewald
A class K of rings has the GADS property (i.e., generalized ADS property) if wheneverX& I& R with X∈ K, then there exists B & R with B ∈ K such that X ⊆ B ⊆ I. Radicals whose semisimple classes have the GADS property are called g-radicals. In this paper, we fully characterize the class of g -radicals. We show that ? is a g-radical if and only if either ? ⊆ I or S? ⊆ I , whereI denotes the class of idempotent rings and S? denotes the semisimple class of ?. It is also shown that the (hereditary) g-radicals form an (atomic) sublattice of the lattice of all radicals.