Weakly continuous and C2-rings

A ring R is called right weakly continuous if the right annihilator of each element is essential in a summand of R, and R satisfies the right C2-condi tion (every right ideal that is isomorphic to a direct summand of R is itse lf a direct summand). We show that a ring R is right weakly continuous if a nd only if it is semiregular and J(R) = Z(R-R). Unlike right continuous rin gs, these right weakly continuous rings form a Morita invariant class. The rings satisfying the right C2-condition are studied and used to investigate two conjectures about strongly right Johns rings and right FGF-rings and t heir relation to quasi-Frobenius rings.