ON RADICALS OF LEFT V-SEMIRINGS
Abstract
In this paper, we solve Problem 1 in [8] for left V-semirings. Specifically,we prove that has the inclusion of the radical J_s(R) into the Jacobson radical
J (R) for every left V-semiring R. Moreover, we give a necessary and sufficient
condition for two radicals are equal on left artinian (or subtractive) left V-semirings.
References
J. Y. Abuhlail, S. N. Il'in, Y. Katsov and T. G. Nam, On V-Semirings and Semirings all of whose Cyclic Semimodules are Injective, Accepted in Comm. Algebra (Also, see, arXiv: 1406.0590v1).
S. Bourne, The Jacobson radical of a semiring, Proc. Nat. Acad. Sci. USA 37(1951), 163-170.
J. Gardner and R. Wiegandt, Radical Theory of Rings, Marcel Dekker, Inc., New York, Basel, 2004.
J. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht-Boston-London, 1999.
S. N. Il'in, V -semirings, Siberian Math. J. 53, 2(2012), 222-231.
K. Iizuka, On the Jacobson radical of a semiring, Tohoku Math. J. 2(1959), 409 - 421.
Y. Katsov, T. G. Nam and N. X. Tuyen, On subtractive semisimple semirings, Algebra Colloq. 16(2009), 415-426.
Y. Katsov and T. G. Nam, On radicals of semirings and related problems, Comm. Algebra 42(2014), 5065-5099.
T. Y. Lam, Lectures on Modules and Rings, 2nd ed., Springer-Verlag, New York-Berlin, 2001.
T. Y. Lam, A first course in Noncommutative rings, 2nd ed., SpringerVerlag, New York-Berlin, 2001.
B. Morak, On the radical theory for semirings, Contributions to Algebra and Geometry, 40(1999), 533-549.
N. X. Tuyen and L. H. Mai, Hopkins Theorem about Jacobson radical for additively cancellative semirings , Hue Uni. J. Sci. 59(2010), 155-162.