Abstract
We introduce the class of lineal rings, defined by the property that the lattice of right annihilators is linearly ordered. We obtain results on the structure of these rings, their ideals, and important radicals; for instance, we show that the lower and upper nilradicals of these rings coincide. We also obtain an affirmative answer to the Köthe Conjecture for this class of rings. We study the relationships between lineal rings, distributive rings, Bézout rings, strongly prime rings, and Armendariz rings. In particular, we show that lineal rings need not be Armendariz, but they fall not far short.
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S. A. Amitsur, Nil radicals. Historical notes and some new results, in Rings, modules and radicals (Proc. Internat. Colloq., Keszthely, 1971), North-Holland, Amsterdam, 1973, pp. 47–65. Colloq. Math. Soc. János Bolyai, Vol. 6.
D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265–2272.
V. A. Andrunakievič, Radicals of associative rings. I, Mat. Sb. N.S. 44(86) (1958), 179–212.
E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470–473.
G. M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), 178–218.
C. Bessenrodt, H. H. Brungs and G. Törner, Right chain rings, part 1, Schriftenreihe des Fachbereichs Mathematik der Universität Duisburg, Vol. 181, 1990.
H. H. Brungs, Rings with a distributive lattice of right ideals, J. Algebra 40 (1976), 392–400.
H. H. Brungs and N. I. Dubrovin, A classification and examples of rank one chain domains, Trans. Amer. Math. Soc. 355 (2003), 2733–2753.
V. Camillo, Distributive modules, J. Algebra 36 (1975), 16–25.
V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), 599–615.
M. Ferrero and G. Törner, On the ideal structure of right distributive rings, Comm. Algebra 21 (1993), 2697–2713.
K. R. Goodearl and D. Handelman, Simple self-injective rings, Comm. Algebra 3 (1975), 797–834.
D. Handelman and J. Lawrence, Strongly prime rings, Trans. Amer. Math. Soc. 211 (1975), 209–223.
Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), 45–52.
Y. Hirano, D. van Huynh and J. K. Park, On rings whose prime radical contains all nilpotent elements of index two, Arch. Math. (Basel) 66 (1996), 360–365.
C. Huh, Y. Lee and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), 751–761.
C. U. Jensen, On characterizations of Prüfer rings, Math. Scand. 13 (1963), 90–98.
N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477–488.
T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, Vol. 189, Springer-Verlag, New York, 1999.
T. Y. Lam, A first course in noncommutative rings, second ed., Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, New York, 2001.
T.-K. Lee and T.-L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), 583–593 (electronic).
G. Marks and R. Mazurek, Annelidan rings, Forum Math., to appear.
G. Marks, R. Mazurek and M. Ziembowski, A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010), 361–397.
R. Mazurek, Distributive rings with Goldie dimension one, Comm. Algebra 19 (1991), 931–944.
R. Mazurek and E. R. Puczyłowski, On nilpotent elements of distributive rings, Comm. Algebra 18 (1990), 463–471.
E. R. Puczyłowski, Questions related to Koethe’s nil ideal problem, in Algebra and its applications, Contemp. Math., Vol. 419, Amer. Math. Soc., Providence, RI, 2006, pp. 269–283.
R. A. Rubin, Absolutely torsion-free rings, Pacific J. Math. 46 (1973), 503–514.
W. Stephenson, Modules whose lattice of submodules is distributive, Proc. London Math. Soc. (3) 28 (1974), 291–310.
A. A. Tuganbaev, Distributive rings, uniserial rings of fractions, and endo-Bezout modules, J. Math. Sci. (N. Y.) 114 (2003), 1185–1203, Algebra, 22.
P. Vámos, Finitely generated Artinian and distributive modules are cyclic, Bull. London Math. Soc. 10 (1978), 287–288.
H.-P. Yu, On quasi-duo rings, Glasgow Math. J. 37 (1995), 21–31.
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Marks, G., Mazurek, R. Rings with linearly ordered right annihilators. Isr. J. Math. 216, 415–440 (2016). https://doi.org/10.1007/s11856-016-1415-5
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DOI: https://doi.org/10.1007/s11856-016-1415-5