On the Vertices of Modules in the Auslander-Reiten Quiver of p-Groups.
Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...
A ring is called right P-injective if every homomorphism from a principal right ideal of to can be extended to a homomorphism from to . Let be a ring and a group. Based on a result of Nicholson and Yousif, we prove that the group ring is right P-injective if and only if (a) is right P-injective; (b) is locally finite; and (c) for any finite subgroup of and any principal right ideal of , if , then there exists such that . Similarly, we also obtain equivalent characterizations...
In this paper we study restricted Boolean rings and group rings. A ring is if every proper homomorphic image of is boolean. Our main aim is to characterize restricted Boolean group rings. A complete characterization of non-prime restricted Boolean group rings has been obtained. Also in case of prime group rings necessary conditions have been obtained for a group ring to be restricted Boolean. A counterexample is given to show that these conditions are not sufficient.
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ; where is a Boolean ring; local ring with nil Jacobson radical; or ; or the ring of a Morita context with zero pairings where the underlying rings are or .