×

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)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Anderson F. W., Second Edition, in: Rings and Categories of Modules (1992) · Zbl 0765.16001
[2] DOI: 10.1515/crll.1970.245.63 · Zbl 0211.36401
[3] Björk J.-E., J. Reine Angew. Math. 253 pp 78– (1972) · Zbl 0228.16011
[4] DOI: 10.4153/CMB-1991-074-x · Zbl 0767.16004
[5] DOI: 10.1007/BF02761306 · Zbl 0802.16010
[6] DOI: 10.1080/00927878608823362 · Zbl 0596.16015
[7] DOI: 10.1090/S0002-9939-1992-1100651-0
[8] Faith C., Proc. A.M.S. 120 pp 1071– (1994)
[9] DOI: 10.1090/S0002-9939-97-03747-7 · Zbl 0871.16012
[10] Goodearl K. R., Ring Theory: Nonsingular Rings and Modules (1976)
[11] Kasch F., Modules and Rings (1982) · Zbl 0523.16001
[12] DOI: 10.2748/tmj/1178243180 · Zbl 0175.31802
[13] DOI: 10.4153/CJM-1976-109-2 · Zbl 0317.16005
[14] DOI: 10.1017/S0017089500032535 · Zbl 0910.16001
[15] DOI: 10.1006/jabr.1995.1117 · Zbl 0839.16004
[16] DOI: 10.1006/jabr.1996.6796 · Zbl 0879.16002
[17] DOI: 10.1016/0021-8693(66)90028-7 · Zbl 0171.29303
[18] Small L., Bull. A.M.S. 73 pp 656– (1967) · Zbl 0149.28102
[19] DOI: 10.1090/S0002-9947-1965-0174592-8
[20] DOI: 10.1006/jabr.1996.6936 · Zbl 0884.16002
[21] DOI: 10.1080/00927879808826127 · Zbl 0893.16010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.