Hong, Chan Yong; Kim, Nam Kyun; Kwak, Tai Keun Ore extensions of Baer and p.p.-rings. (English) Zbl 0982.16021 J. Pure Appl. Algebra 151, No. 3, 215-226 (2000). A ring \(R\) is called Baer (resp. right PP) if the right annihilator of every nonempty subset (resp. singleton subset) is generated as a right ideal by an idempotent. Similarly left PP rings can be defined. A ring is called PP if it is both right and left PP. Let \(\alpha\) be an endomorphism of a ring \(R\). Then according to J. Krempa [Algebra Colloq. 3, No. 4, 289-300 (1996; Zbl 0859.16019)] \(\alpha\) is said to be rigid if \(r\alpha(r)=0\) implies \(r=0\) for \(r\in R\). A ring \(R\) is said to be \(\alpha\)-rigid if there exists a rigid endomorphism \(\alpha\) of \(R\). For an endomorphism \(\alpha\) and \(\delta\) an \(\alpha\)-derivation of a ring \(R\), assume that \(R\) is \(\alpha\)-rigid. Then it is shown that \(R\) is a Baer ring if and only if the Ore extension \(R[x;\alpha,\delta]\) is a Baer ring if and only if the skew power series ring \(R[[x;\alpha]]\) is a Baer ring. Also, it is proved under the same hypothesis that \(R\) is a PP ring if and only if the Ore extension \(R[x;\alpha,\delta]\) is a PP ring. Reviewer: J.K.Park (Pusan) Cited in 4 ReviewsCited in 105 Documents MSC: 16S36 Ordinary and skew polynomial rings and semigroup rings 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) Keywords:right annihilators; idempotents; left PP rings; rigid endomorphisms; Baer rings; Ore extensions; skew power series rings Citations:Zbl 0859.16019 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Armendariz, E. P., A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc., 18, 470-473 (1974) · Zbl 0292.16009 [2] Armendariz, E. P.; Koo, H. K.; Park, J. K., Isomorphic Ore extensions, Comm. Algebra, 15, 12, 2633-2652 (1987) · Zbl 0629.16002 [3] Birkenmeier, G. F., Baer rings and quasi-continuous rings have a MDSN, Pacific J. Math., 97, 283-292 (1981) · Zbl 0432.16010 [4] Birkenmeier, G. F., Idempotents and completely semiprime ideals, Comm. Algebra, 11, 567-580 (1983) · Zbl 0505.16004 [5] Birkenmeier, G. F., Decompositions of Baer-like rings, Acta Math. Hungar., 59, 319-326 (1992) · Zbl 0771.16003 [6] G.F. Birkenmeier, J.Y. Kim, J.K. Park, On extensions of quasi-Baer and principally quasi-Baer rings, Preprint.; G.F. Birkenmeier, J.Y. Kim, J.K. Park, On extensions of quasi-Baer and principally quasi-Baer rings, Preprint. · Zbl 0987.16017 [7] Chase, S., A generalization of the ring of triangular matrices, Nagoya Math. J., 18, 13-25 (1961) · Zbl 0113.02901 [8] A.W. Chatters, C.R. Hajarnavis, Rings with Chain Conditions, Pitman, Boston, 1980.; A.W. Chatters, C.R. Hajarnavis, Rings with Chain Conditions, Pitman, Boston, 1980. · Zbl 0446.16001 [9] Clark, W. E., Twisted matrix units semigroup algebras, Duke Math. J., 34, 417-424 (1967) · Zbl 0204.04502 [10] Y. Hirano, On isomorphisms between Ore extensions, Preprint.; Y. Hirano, On isomorphisms between Ore extensions, Preprint. [11] Jøndrup, S., p.p. rings and finitely generated flat ideals, Proc. Amer. Math. Soc., 28, 431-435 (1971) · Zbl 0195.32703 [12] Kamal, A. M., Some remarks on Ore extension rings, Comm. Algebra, 22, 3637-3667 (1994) · Zbl 0822.16026 [13] I. Kaplansky, Rings of Operators, Math. Lecture Note Series, Benjamin, New York, 1965.; I. Kaplansky, Rings of Operators, Math. Lecture Note Series, Benjamin, New York, 1965. · Zbl 0174.18503 [14] Krempa, J., Some examples of reduced rings, Algebra Colloq., 3, 4, 289-300 (1996) · Zbl 0859.16019 [15] Pollingher, A.; Zaks, A., On Baer and quasi-Baer rings, Duke Math. J., 37, 127-138 (1970) · Zbl 0219.16010 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. 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.