ON THE LEVITZKI RADICAL OF MODULES

Abstract

In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we define and characterize l-prime submodules. Let N be a submodule of an R-module M. If l.√N := {m ∈ M : every l- system of M containingm meets N}, we show that l.√N coincides with the intersection L(N) of all l-prime submodules of M containing N. We define the Levitzki radical of an R-module M as L(M) = l.√0. Let β(M), U(M) and Rad(M) be the prime radical, upper nil radical and Jacobson radical of M respectively. In general β(M) ⊆ L(M) ⊆ U(M) ⊆ Rad(M). If R is commutative, β(M) = L(M) = U(M) and if R is left Artinian, β(M) = L(M) = U(M) = Rad(M). Lastly, we show that the class of all l-prime modules RM with RM 6= 0 forms a special class of modules.

Keywords

• l-prime submodule, semi l-prime, s-prime submodule, upper nil radical, Levitzki radical

References

1. V. A Andrunakievich and Ju M. Rjabuhin, Special modules and special radicals, Soviet Math. Dokl., 3 (1962), 1790–1793. Russian original: Dokl. Akad. Nauk SSSR., 147 (1962), 1274–1277.
2. A. M. Babic, Levitzki radical, Doklady Akad. Nauk. SSSR., 126 (1950), 242-243 (Russian).
3. M. Behboodi, A generalization of Baer’s lower nilradical for modules, J. Alge- bra Appl., 6 (2007), 337–353.
4. M. Behboodi, On the prime radical and Baer’s lower nilradical of modules, Acta Math. Hungar., 122 (2008), 293–306.
5. L. Bican, T. Kepka and P. Nemec, Rings, modules and preradicals, Lecture Notes in Pure and Applied Mathematics no.75, Marcel Dekker Inc., New York, J. Dauns, Prime modules, J. Reine. Angew. Math., 298 (1978), 156–181.
6. B. De La Rosa and S. Veldsman, A relationship between ring radicals and module radicals, Quaest. Math., 17 (1994), 453–467.
7. B. J. Gardner and R. Wiegandt, Radical Theory of Rings, New York: Marcel Dekker, 2004.
8. N. J. Groenewald and P. C. Potgieter, A note on the Levitzki radical of a near-ring, J. Austral. Math. Soc. (Series A), 36 (1984), 416-420.

9. N. J. Groenewald and D. Ssevviiri, K¨othe upper nilradical for modules, Acta Math. Hungar., 138(4) (2013), 295–306.
10. N. J. Groenewald and D. Ssevviiri, 2-primal modules, J. Algebra Appl., 12(5) (2013), DOI:10.1142/S021949881250226X.
11. J. Jenkins and P. F. Smith, On the prime radical of a module over a commu- tative ring, Comm. Algebra, 20 (1992), 3593–3602.
12. W. K. Nicholson and J. F. Watters, The strongly prime radical, Proc. Amer. Math. Soc., 76 (1979), 235–240.
13. L. H. Rowen, Ring theory, Academic Press, Inc., San Diego, 1991.
14. D. Ssevviiri, A contribution to the theory of prime modules, PhD Thesis, Nelson Mandela Metropolitan University, 2012.
15. A. P. J. Van der Walt, Contributions to ideal theory in general rings, Proc. Kon. Ned. Akad. Wetensch., Ser. A., 67 (1964), 68–77.
16. A. P. J. Van der Walt, On the Levitzki nil radical, Archiv Math., XVI (1965), 24.
17. R. Wiegandt, Rings decisive in radical theory, Quaest. Math., 22 (1999), 303- Nico J. Groenewald
18. Department of Mathematics and Applied Mathematics Nelson Mandela Metropolitan University Port Elizabeth South Africa e-mail: [e-mail] David Ssevviiri Department of Mathematics and Applied Mathematics Nelson Mandela Metropolitan University Port Elizabeth South Africa e-mail: [e-mail]