×

zbMATH — the first resource for mathematics

Triangular matrix representations of ring extensions. (English) Zbl 1054.16018
From the authors’ abstract: We investigate the class of piecewise prime, PWP, rings which properly includes all piecewise domains (hence all right hereditary rings which are semiprimary or right Noetherian). For a PWP ring we determine a large class of ring extensions which have a generalized triangular matrix representation for which the diagonal rings are prime.

MSC:
16S20 Centralizing and normalizing extensions
16S50 Endomorphism rings; matrix rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16N60 Prime and semiprime associative rings
20M25 Semigroup rings, multiplicative semigroups of rings
PDF BibTeX XML Cite
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] E.P. Armendariz, H.K. Koo, J.K. Park, Ore extensions of von Neumann regular rings, Preprint
[3] Behn, A., Polycyclic group rings whose principal ideals are projective, J. algebra, 232, 697-707, (2000) · Zbl 0972.16013
[4] Berberian, S.K., Baer \(∗\)-rings, (1972), Springer-Verlag Berlin-Heidelberg-New York
[5] Birkenmeier, G.F., Baer rings and quasi-continuous rings have a MDSN, Pacific J. math., 97, 283-292, (1981) · Zbl 0432.16010
[6] Birkenmeier, G.F., Idempotents and completely semiprime ideals, Comm. algebra, 11, 567-580, (1983) · Zbl 0505.16004
[7] Birkenmeier, G.F., Decompositions of Baer-like rings, Acta math. hungar., 59, 319-326, (1992) · Zbl 0771.16003
[8] Birkenmeier, G.F.; Heatherly, H.E.; Kim, J.Y.; Park, J.K., Triangular matrix representations, J. algebra, 230, 558-595, (2000) · Zbl 0964.16031
[9] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., Quasi-Baer ring extensions and biregular rings, Bull. austral. math. soc., 61, 39-52, (2000) · Zbl 0952.16009
[10] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., A sheaf representation of quasi-Baer rings, J. pure appl. algebra, 146, 209-223, (2000) · Zbl 0947.16018
[11] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., On quasi-Baer rings, (), 67-92 · Zbl 0974.16006
[12] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., On polynomial extensions of principally quasi-Baer rings, Kyungpook math. J., 40, 247-253, (2000) · Zbl 0987.16017
[13] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., Principally quasi-Baer rings, Comm. algebra, 29, 639-660, (2001) · Zbl 0991.16005
[14] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., Semicentral reduced algebras, (), 67-84 · Zbl 0987.16010
[15] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., Polynomial extensions of Baer and quasi-Baer rings, J. pure appl. algebra, 159, 25-42, (2001) · Zbl 0987.16018
[16] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., Triangular matrix representations of semiprimary rings, J. algebra and its appl., 1, 123-131, (2002) · Zbl 1034.16032
[17] Birkenmeier, G.F.; Kim, J.Y.; Park, J.K., Prime ideals of principally quasi-Baer rings, Acta math. hungar., 98, 217-225, (2003) · Zbl 1066.16018
[18] Brown, K.A., The singular ideals of group rings, Quart. J. math. Oxford, 28, 41-60, (1977) · Zbl 0345.16012
[19] Camillo, V.P., Semihereditary polynomial rings, Proc. amer. math. soc., 45, 173-174, (1974) · Zbl 0294.13014
[20] Clark, W.E., Twisted matrix units semigroup algebras, Duke math. J., 34, 417-424, (1967) · Zbl 0204.04502
[21] Gordon, R.; Small, L.W., Piecewise domains, J. algebra, 23, 553-564, (1972) · Zbl 0244.16008
[22] Groenewald, N., A note on extensions of bear and p.p.-rings, Publ. L’institute math., 34, 71-72, (1983) · Zbl 0549.20051
[23] Hirano, Y., On ordered monoid rings over a quasi-Baer ring, Comm. algebra, 29, 2089-2095, (2001) · Zbl 0996.16020
[24] Jøndrup, S., P.p. rings and finitely generated flat ideals, Proc. amer. math. soc., 28, 431-435, (1971) · Zbl 0195.32703
[25] Kaplansky, I., Rings of operators, (1965), Benjamin New York
[26] Krempa, J.; Niewieczerzal, D., Rings in which annihilators are ideals and their application to semigroup rings, Bull. acad. polon. sci. math and astronom. phys., 25, 851-856, (1977) · Zbl 0345.16017
[27] Lam, T.Y., Lectures on modules and rings, (1998), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0911.16001
[28] McCarthy, P.J., The ring of polynomials over a commutative von Neumann regular ring, Proc. amer. math. soc., 39, 253-254, (1973) · Zbl 0267.13008
[29] Okniński, J., Semigroup algebras, (1991), Marcel Dekker New York · Zbl 1178.16025
[30] Passman, D.S., The algebraic structure of group rings, (1977), Wiley New York · Zbl 0366.16003
[31] Pillay, P., On semihereditary noncommutative polynomial rings, Proc. amer. math. soc., 78, 473-474, (1980) · Zbl 0446.16012
[32] Pollingher, A.; Zaks, A., On Baer and quasi-Baer rings, Duke math. J., 37, 127-138, (1970) · Zbl 0219.16010
[33] Small, L.W., Semi-hereditary rings, Bull. amer. math. soc., 73, 656-658, (1967) · Zbl 0149.28102
[34] Smith, M.K., Group algebras, J. algebra, 18, 477-499, (1971) · Zbl 0219.20002
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.