## 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.

### MSC:

 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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### References:

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