×

On a problem of Bertram Yood. (English) Zbl 1294.16034

The paper contains a partial answer to the question of B. Yood [posed in Can. J. Math. 16, 28-45 (1964; Zbl 0203.04805)]: Is it true that in any associative topological ring the intersection \(\mathcal M_l(R)\) of all closed maximal regular left ideals coincides with the intersection \(\mathcal M_r(R)\) of all closed maximal regular right ideals of \(R\)?
An element \(r\) of an associative topological ring is called left topologically quasi-invertible if there is a net \((r_\lambda)_{\lambda\in\Lambda}\) such that the net \((r_\lambda+r-r_\lambda r)_{\lambda\in\Lambda}\) converges to zero. The authors prove that \(\mathcal M_l(R)=\mathcal M_r(R)\) for any associative topological ring \(R\) satisfying the following two conditions: (i) every left (right) topologically quasi-invertible element is left (right) quasi-invertible; (ii) every closed regular left (right) ideal is contained in a closed left (right) maximal ideal.

MSC:

16W80 Topological and ordered rings and modules
16D25 Ideals in associative algebras
46H05 General theory of topological algebras
16N80 General radicals and associative rings

Citations:

Zbl 0203.04805
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mati Abel, Advertive topological algebras. General topological algebras (Tartu, 1999), 14-24, Math. Stud. (Tartu) 1, Est. Math. Soc., Tartu, 2001.; · Zbl 1044.46038
[2] Mati Abel, Descriptions of the topological radical in topological algebras. General topological algebras (Tartu, 1999), 25-31, Math. Stud. (Tartu) 1, Est. Math. Soc., Tartu, 2001.; · Zbl 1044.46039
[3] V.K. Balachandran, Topological Algebras. North-Holland Math. Studies 185, Elsevier, Amsterdam, 2000.;
[4] I. Kaplansky, Topological rings, Amer. J. Math. 69 (1947), 153-183.; · Zbl 0034.16604
[5] I. Kaplansky, Locally compact rings II, Amer. J. Math. 73 (1951), 20-24.; · Zbl 0045.16102
[6] T.Y. Lam, A first course in noncommutative rings. Graduate Texts in Math. 131 (2. ed.), Springer-Verlag, New York, 2001.; · Zbl 0980.16001
[7] E.L. Lody, The Jacobson radical of rings in http://www.math.hawaii.edu/ lee/algebra/radical.pdf;
[8] S. Warner, Topological rings, North-Holland Math. Studies 178, North-Holland Publ. Co., Amsterdam, 1993.;
[9] B. Yood, Ideals in topological rings, Canad. J. Math. 16 (1964), 24-45.; · Zbl 0203.04805
[10] B. Yood, Incomplete normed algebras, Bull. Amer. Math. Soc. 78 (1972), 50-52.; · Zbl 0232.46042
[11] W. Zelazko, Selected topics in topological algebras. Lect. Notes Ser. 31, Aarhus Univ., 1971.; · Zbl 0221.46041
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.