We determine some classes of left modules satisfying the radical formula in a noncommutative ring. We also show that, under a certain condition, a finitely generated module over an HNP -ring (a generalization of the Dedekind domain) both satisfies the radical formula and can be decomposed into a direct sum of torsion and extending modules.
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References
M. Alkan and Y. Tıraş, “On prime submodules,” Rocky Mountain J. Math., 37, No. 3, 709–722 (2007).
M. Alkan and Y. Tıraş, “Projective modules and prime submodules,” Czechoslovak Math. J., 56, No. 2, 601–611 (2006).
F. Ḉallıalp and Ü. Tekir, “On the prime radical of a module over a noncommutative ring,” Taiwanese J. Math., 8, No. 2, 337–341 (2004).
S. Ḉeken and M. Alkan, “On prime submodules and primary decompositions in two-generated free modules,” Taiwan. J. Math., 17, 133–142 (2013).
S. Ḉeken and M. Alkan, “On 𝜏-extending modules,” Mediterran. J. Math., 9, 129–142 (2012).
T. Y. Lam, A First Course in Noncommutative Rings, Springer (2001).
J. C. McConnel and J. C. Robson, Noncommutative Noetherian Rings, Wiley, Chichester (1987).
P. F. Smith, “Radical submodules and uniform dimension of modules,” Turkish J. Math., 28, 255–270 (2004).
J. Dauns, “Prime modules and one-sided ideals in ring theory and algebra,” Ring Theory and Algebra, III : Proc. of the 3rd Oklahoma Conf.,” M. Dekker, New York (1980), pp. 301–344.
J. Jenkins and P. F. Smith, “On the prime radical of a module over a commutative ring,” Comm. Algebra, 20, No. 12, 3593–3602 (1992).
P. F. Smith, “Primary modules over commutative rings,” Glasgow Math. J., 43, 103–111 (2001).
N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, “Extending modules,” Pitman Research Notes in Mathematics No. 313, Longman Scientific & Technical, Harlow (1994).
R. L. McCasland and P. F. Smith, “Prime submodules of Noetherian modules,” Rocky Mountain J. Math., 23, 1041–1062 (1993).
R. L. McCasland and M. E. Moore, “On radicals of submodules,” Comm. Algebra, 19, 1327–1341 (1991).
D. Pusat-Yılmaz and P. F. Smith, “Modules which satisfy the radical formula,” Acta Math. Hungar., 95, No. 1-2, 155–167 (2002).
Y. Tıraş and M. Alkan, Prime modules and submodules,” Comm. Algebra, 31, 395–396 (2003).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 9, pp. 1241–1248, September, 2019.
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Öneş, O., Alkan, M. The Radical Formula for Noncommutative Rings. Ukr Math J 71, 1419–1428 (2020). https://doi.org/10.1007/s11253-020-01723-y
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DOI: https://doi.org/10.1007/s11253-020-01723-y