Jonah, David Rings with the minimum condition for principal right ideals have the maximum condition for principal left ideals. (English) Zbl 0213.04303 Math. Z. 113, 106-112 (1970). Studiert man z.B. Goldie-Ringe, so ist stets nützlich, wenn auch trivial, daß die Minimumbedingung für Linksannullatoren äquivalent ist zur Maximumbedingung für Rechtsannullatoren. Die analoge Aussage des Titels für Hauptideale wird jedoch recht kunstvoll bewiesen, indem interessante Ringtheorie herangezogen wird. Insbesondere folgt die Aussage, nachdem der Autor gezeigt hat, daß ein Ring \( R \) genau dann perfekt ist, wenn jeder monogene R-L1nksmodul die Maximumbedingung besitzt. Denn H. B a s s [Trans. Amer. math. Soc. 95, 466-488 (1960, dies. Zbl. 94, 22)] hat \( \mathrm{R} \) perfekt genannt, wenn es auf jeden \( \mathrm{R}-\mathrm{L} \) inksmodul einen sog. mintmalen Epimorphismus eines projektiven Moduls gibt. Er zeigte dann nämlich, da \( \beta \) \( \mathrm{R} \) perfekt äquivalent ist mit: \( \mathrm{R} \) besitzt die Minimalbedingung für Hauptrechtsideale. This review text is based on a scanned copy of the printed version. It was converted to LaTeX using MathPix and a specifically developed LLM to assign the text parts to the metadata. It may contain errors, misassignments, or gaps; in particular, the reviewer signature has not yet been extracted reliably in general. If you notice any errors, please report them directly to our editorial team via the Contact Form. Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 24 Documents MSC: 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16D25 Ideals in associative algebras × Cite Format Result Cite Review PDF Full Text: DOI EuDML Geodesic References: [1] Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc.95, 466-488 (1960). · Zbl 0094.02201 · doi:10.1090/S0002-9947-1960-0157984-8 [2] Faith, C.: Rings with minimum condition on principal ideals. Arch. der Math.10, 327-330 (1959) [cf. remarks added in proof]. · Zbl 0098.03002 · doi:10.1007/BF01240806 [3] Fuchs, L., Rangaswamy, K. M.: On generalized regular rings. Math. Z.107, 71-81 (1968). · Zbl 0167.03401 · doi:10.1007/BF01111051 [4] Kaplansky, I.: Topological representation of algebras. II. Trans. Amer. Math. Soc.68, 62-75 (1950) [cf. Section 8]. · Zbl 0035.30301 · doi:10.1090/S0002-9947-1950-0032612-4 [5] McCoy, N. H.: Generalized regular rings. Bull. Amer. Math. Soc.45, 175-178 (1939). · Zbl 0020.20001 · doi:10.1090/S0002-9904-1939-06933-4 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.