Weakly continuous and \(C2\)-rings. (English) Zbl 0983.16002

All rings are associative with unity and all \(R\)-modules are unital. Semiregular rings (that is, rings \(R\) such that \(R/J(R)\) is regular and idempotents lift modulo \(J(R)\)) are studied first. It is proved that a ring \(R\) is semiregular with \(J(R)=Z(R_R)\) if and only if the right annihilator of every element is essential in a direct summand of \(R\), and every right ideal that is isomorphic to a direct summand of \(R\) is itself a summand (\(R\) is a right \(C2\)-ring). A ring satisfying these conditions is called a right weakly continuous ring and it is shown that \(R\) is right weakly continuous if and only if it is \(Z(R_R)\)-semiregular. It is also shown that right weak continuity is a generalization of continuity, but that, unlike right continuity, right weak continuity is a Morita invariant property of rings.
\(C2\)-rings are subsequently studied and relationships are found between a module being \(C2\) and its endomorphism ring being \(C2\). In the final section these results are used to study two questions concerning quasi-Frobenius rings. The FGF-conjecture asserts that every FGF-ring is quasi-Frobenius. It is shown that the condition that every right FGF-ring is QF is equivalent to every right FGF-ring being right \(C2\), thus simplifying what is required to prove the conjecture. Another open question is whether strongly right Johns rings are QF. The paper concludes with a proof that strongly right Johns, right \(C2\)-rings are QF.


16D50 Injective modules, self-injective associative rings
16L60 Quasi-Frobenius rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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