Phantom Ideas And Essential Cotorsion Envelopes Over Rings And Algebras

In this paper, we consider the phantom ideal over rings and algebras. It is shown that if two group algebra is stable equivalent, then they have the same phantom number. As applications, we give two examples to show that one can sharp the upper bound of the phantom numbers of group algebras by this method. We then show that for an IF ring, a map is phantom if and only if it is cophantom. We also study the rings with right modules having essential cotorsion envelopes. Let R a ring with identity. It is shown that every right R-module has an essential cotorsion envelope if and only if the class of singular and cyclic flat right R-modules L is closed under quotients, and the class of cotorsion right R-modules C is the right orthogonal class of L. As an application, we obtain that if R is commutative ring, and every R-module has an essential cotorsion envelope, then R is a max ring, i.e., R/J(R) is von Neuman regular, and J(R) is T-nilpotent, where J(R) is the Jacobson radical.