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On Annelidan, Distributive, and Bézout Rings

Part of: Modules, bimodules and ideals Conditions on elements Radicals and radical properties of rings Distributive lattices Chain conditions, growth conditions, and other forms of finiteness Local rings and generalizations Rings and algebras arising under various constructions Boolean algebras (Boolean rings)

Published online by Cambridge University Press:  03 May 2019

Greg Marks  and
Ryszard Mazurek
Greg Marks
Affiliation:
Department of Mathematics and Statistics, St. Louis University, St. Louis, MO63103, USA Email: [email protected]
Ryszard Mazurek
Affiliation:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351Białystok, Poland Email: [email protected]
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Abstract

A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.

Keywords

annelidan ringdistributive ringBézout ring

MSC classification

Primary: 06D99: None of the above, but in this section
Secondary: 16D25: Ideals 16P50: Localization and Noetherian rings 16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence 16S85: Rings of fractions and localizations 16U20: Ore rings, multiplicative sets, Ore localization 16L30: Noncommutative local and semilocal rings, perfect rings 16N20: Jacobson radical, quasimultiplication 16N60: Prime and semiprime rings 16P40: Noetherian rings and modules 16P60: Chain conditions on annihilators and summands
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 1082 - 1110
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The second author was supported by Polish KBN Grant 1 P03A 032 27. Parts of this paper were written while the first author was visiting the Bialystok University of Technology; other parts were written while the second author was visiting St. Louis University. Each is deeply grateful for the warm hospitality extended by both institutions.

References

Auslander, M., Green, E. L., and Reiten, I., Modules with waists. Illinois J. Math. 19(1975), 467–478.CrossRefGoogle Scholar
Badawi, A., Pseudo-valuation domains: a survey. In: Mathematics & mathematics education (Bethlehem, 2000). World Sci. Publ., River Edge, NJ, 2002, pp. 38–59.CrossRefGoogle Scholar
Bell, J., Rogalski, D., and Sierra, S. J., The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings. Israel J. Math. 180(2010), 461–507. https://doi.org/10.1007/s11856-010-0111-0CrossRefGoogle Scholar
Bessenrodt, C., Brungs, H.-H., and Törner, G., Prime ideals in right chain rings. Mitt. Math. Sem. Giessen(1984), no. 163, 141–167.Google Scholar
Bessenrodt, C., Brungs, H. H., and Törner, G., Right chain rings, part 1. Schriftenreihe des Fachbereichs Mathematik der Universität Duisburg, 1990, vol. 181.Google Scholar
Bourbaki, N., Elements of mathematics. Commutative algebra. Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972.Google Scholar
Brungs, H. H. and Dubrovin, N. I., A classification and examples of rank one chain domains. Trans. Amer. Math. Soc. 355(2003), no. 7, 2733–2753. https://doi.org/10.1090/S0002-9947-03-03272-0CrossRefGoogle Scholar
Cohn, P. M., Free ideal rings and localization in general rings. New Mathematical Monographs, 3, Cambridge University Press, Cambridge, 2006. https://doi.org/10.1017/CBO9780511542794CrossRefGoogle Scholar
Dubrovin, N. I., The rational closure of group rings of left-ordered groups. translated from Mat. Sb. 184(1993), no. 7, 3–48. Russian Acad. Sci. Sb. Math. 79(1994), no. 2, 231–263. https://doi.org/10.1070/SM1994v079n02ABEH003498Google Scholar
Ferrero, M. and Mazurek, R., On the structure of distributive and Bezout rings with waists. Forum Math. 17(2005), no. 2, 191–198. https://doi.org/10.1515/form.2005.17.2.191CrossRefGoogle Scholar
Ferrero, M. and Sant’Ana, A., Rings with comparability. Canad. Math. Bull. 42(1999), no. 2, 174–183. https://doi.org/10.4153/CMB-1999-021-xCrossRefGoogle Scholar
Ferrero, M. and Törner, G., On the ideal structure of right distributive rings. Comm. Algebra 21(1993), 8, 2697–2713. https://doi.org/10.1080/00927879308824701CrossRefGoogle Scholar
Ghashghaei, E., Koşan, M. T., Namdari, M., and Yildirim, T., Rings in which every left zero-divisor is also a right zero-divisor and conversely. J. Algebra Appl. 18(2019), no. 5, 1950096. https://doi.org/10.1142/S0219498819500968CrossRefGoogle Scholar
Goel, V. K. and Jain, S. K., 𝜋-injective modules and rings whose cyclics are 𝜋-injective. Comm. Algebra 6(1978), no. 1, 59–73. https://doi.org/10.1080/00927877808822233CrossRefGoogle Scholar
Goodearl, K. R., von Neumann regular rings. Second ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.Google Scholar
Goodearl, K. R. and Warfield, R. B. Jr., An introduction to noncommutative Noetherian rings, Second ed., London Mathematical Society Student Texts, 61, Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511841699CrossRefGoogle Scholar
Grams, A., Atomic rings and the ascending chain condition for principal ideals. Proc. Camb. Phil. Soc. 75(1974), 321–329. https://doi.org/10.1017/s0305004100048532CrossRefGoogle Scholar
Hedstrom, J. R. and Houston, E. G., Pseudo-valuation domains. Pacific J. Math. 75(1978), no. 1, 137–147.CrossRefGoogle Scholar
Hedstrom, J. R. and Houston, E. G., Pseudo-valuation domains. II. Houston J. Math. 4(1978), no. 2, 199–207.Google Scholar
Kaplansky, I., Elementary divisors and modules. Trans. Amer. Math. Soc. 66(1949), 464–491. https://doi.org/10.2307/1990591CrossRefGoogle Scholar
Lam, T. Y., Lectures on modules and rings. Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4612-0525-8CrossRefGoogle Scholar
Lam, T. Y., A first course in noncommutative rings, Second ed., Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4419-8616-0CrossRefGoogle Scholar
Lomp, C. and Sant’Ana, A., Comparability, distributivity and non-commutative 𝜙-rings. Groups, rings and group rings. Contemp. Math., 499, Amer. Math. Soc., Providence, RI, 2009, pp. 205–217. https://doi.org/10.1090/conm/499/09804Google Scholar
Marks, G., Duo rings and Ore extensions. J. Algebra 280(2004), no. 2, 463–471. https://doi.org/10.1016/j.jalgebra.2004.04.018CrossRefGoogle Scholar
Marks, G. and Mazurek, R., Annelidan rings. Forum Math. 28(2016), no. 5, 923–941. https://doi.org/10.1515/forum-2015-0107CrossRefGoogle Scholar
Marks, G. and Mazurek, R., Rings with linearly ordered right annihilators. Israel J. Math. 216(2016), no. 1, 415–440. https://doi.org/10.1007/s11856-016-1415-5CrossRefGoogle Scholar
Mazurek, R., Remarks on zero-divisors in chain rings. Arch. Math. (Basel) 52(1989), no. 5, 428–432. https://doi.org/10.1007/BF01198349CrossRefGoogle Scholar
Mazurek, R., Distributive rings with Goldie dimension one. Comm. Algebra 19(1991), no. 3, 931–944. https://doi.org/10.1080/00927879108824179CrossRefGoogle Scholar
Mazurek, R., Pseudo-chain rings and pseudo-uniserial modules. Comm. Algebra 33(2005), no. 5, 1519–1527. https://doi.org/10.1081/AGB-200060527CrossRefGoogle Scholar
Mazurek, R. and Puczyłowski, E. R., On nilpotent elements of distributive rings. Comm. Algebra 18(1990), no. 2, 463–471. https://doi.org/10.1080/00927879008823925CrossRefGoogle Scholar
Mazurek, R. and Törner, G., Comparizer ideals of rings. Comm. Algebra 32(2004), no. 12, 4653–4665. https://doi.org/10.1081/AGB-200036825CrossRefGoogle Scholar
Mazurek, R. and Törner, G., On semiprime segments of rings. J. Aust. Math. Soc. 80(2006), no. 2, 263–272. https://doi.org/10.1017/S1446788700013100CrossRefGoogle Scholar
Puninskaya, V., Modules with few types over a serial ring and over a commutative Prüfer ring. Comm. Algebra 30(2002), no. 3, 1227–1240. https://doi.org/10.1081/AGB-120004870CrossRefGoogle Scholar
Puninskiĭ, G. E. and Tuganbaev, A. A., . In: Izdatel’stvo “SOYuZ”. Moskovskiĭ Gosudarstvennyĭ Sotsial’nyĭ Universitet, Moscow, 1998.Google Scholar
Redmond, S. P., The zero-divisor graph of a non-commutative ring. In: Commutative rings. Nova Sci. Publ., Hauppauge, NY, 2002, 39–47.Google Scholar
Sally, J. D. and Vasconcelos, W. V., Stable rings. J. Pure Appl. Algebra 4(1974), 319–336. https://doi.org/10.1016/0022-4049(74)90012-7CrossRefGoogle Scholar
Schröder, M., Über N. I. Dubrovins Ansatz zur Konstruktion von nicht vollprimen Primidealen in Kettenringen. Results Math. 17(1990), no. 3–4, 296–306. https://doi.org/10.1007/BF03322466CrossRefGoogle Scholar
Shin, G., Prime ideals and sheaf representation of a pseudo symmetric ring. Trans. Amer. Math. Soc. 184(1973), 43–60. https://doi.org/10.2307/1996398CrossRefGoogle Scholar
Sigurđsson, G., Links between prime ideals in differential operator rings. J. Algebra 102(1986), no. 1, 260–283. https://doi.org/10.1016/0021-8693(86)90141-9CrossRefGoogle Scholar
Small, L. W., Prime ideals in Noetherian PI-rings. Bull. Am. Math. Soc. 79(1973), 421–422. https://doi.org/10.1090/S0002-9904-1973-13196-9CrossRefGoogle Scholar
Stephenson, W., Modules whose lattice of submodules is distributive. Proc. Lond. Math. Soc. (3) 28(1974), 291–310. https://doi.org/10.1112/plms/s3-28.2.291CrossRefGoogle Scholar
Törner, G., Left and right associated prime ideals in chain rings with d.c.c. for prime ideals. Results Math. 12(1987), no. 3–4, 428–433. https://doi.org/10.1007/BF03322408CrossRefGoogle Scholar
Tuganbaev, A. A., Distributive rings, uniserial rings of fractions, and endo-Bezout modules. J. Math. Sci. (N. Y.) 114(2003), no. 2, 1185–1203. https://doi.org/10.1023/A:1021977603746CrossRefGoogle Scholar