A contribution to the theory of prime modules

References (124)

1. M fully IFP (e.g., when R is left duo),
2. M is finitely generated (cyclic or free) and R is medial (LSD, left per- mutable or right permutable).
3. 3 Non-nilpotent elements of the Z-module Z/p k Z In this section, we characterize the structure of non-nilpotent elements of the Z-module Z/p k Z. Table 5.3 below gives examples of non-nilpotent elements of the Z-module Z/p k Z. we have (a) ⊆ (P : {am}) such that (a)am ⊆ P , (a)aRm ⊆ P , (a)Ram ⊆ P and (a)RaRm ⊆ P . Hence, (a) 2 m = (a)[Za + aR + Ra + RaR]m ⊆ P . R commutative ⇒ symmetric is trivial. Corollary 6.1.2 For any proper submodule P of an R-module M , the follow- ing statements are equivalent:
4. P is completely semiprime,
5. P is semi-symmetric and classical semiprime,
6. P is semi-symmetric and semiprime,
7. P is IFP and classical semiprime,
8. P is IFP and semiprime,
9. P is symmetric and classical semiprime,
10. P is symmetric and semiprime.
11. Proof: It suffices to prove (2) ⇒ (1), the rest follows from Theorem 6.1.2 and the fact that completely semiprime ⇒ semiprime ⇒ classical semiprime. Suppose for a ∈ R and m ∈ M , a 2 m ∈ P . Then (a) 2 m ⊆ P since P is semi- symmetric. P classical semiprime implies (a)m ⊆ P . Thus, a m ⊆ P and P is completely semiprime. Remark 6.1.1 When P = 0 in Corollary 6.1.2, we get different characteriza- tions of reduced modules. Theorem 6.1.3 For any submodule P of an R-module M , P semi-symmetric submodule ⇒ P is a 2-primal submodule.
12. a reversible ring need not be commutative. Hence, symmetric modules need not be defined over commutative rings. Corollary 6.1.3 Each of the following statements implies that each submodule of a module M is 2-primal: 1. every semiprime submodule of M is completely semiprime, 2. every prime submodule of M contains a completely prime submodule of M , 3. every classical prime submodule of M is completely prime, 4. every prime submodule of M is completely prime. Proof: Elementary.
13. Example 6.1.4 If M is a finitely generated (cyclic or free) module over a medial (left permutable, right permutable or left self distributive) ring, then M is 2-primal. Proof: Follows from Corollary 3.5.2 and Corollary 6.1.3(4). Corollary 6.1.4 For a proper submodule P of an R-module M , the following statements are equivalent:
14. P is a completely prime submodule of M ,
15. P is a prime submodule of M and the module M/P is IFP,
16. P is a prime submodule of M and (P : S) R for all subsets S of M , 6. P is a prime submodule of M and the module M/P is semi-symmetric,
17. P is a prime and 2-primal submodule of M . Proof: 1⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ and 6 ⇒ 7 follow from Theorems 6.1.2 and 6.
18. Suppose P is a prime and 2-primal submodule. Then M/P is a prime module such that β(M/P ) = (0). P 2-primal implies β(M/P ) = β co (M/P ) = (0). Since β co (M/P ) is always a completely semiprime submodule of M/P , M/P is a reduced (equivalently completely semiprime) module. Thus, P is a completely semiprime submodule and hence completely prime. To see this, let am ∈ P for a ∈ R and m ∈ M . P completely semiprime implies P is IF P , hence a m ⊆ P . P prime implies m ∈ P or aM ⊆ P . Thus P is completely prime. Example 6.1.5 Reduced modules are 2-primal.
19. Example 6.1.6 If R is a left duo ring, the module R M is strongly 2-primal, (i.e., every proper submodule of M is 2-primal). Proof: Follows from Example 3.1.1 and Corollary 6.1.
20. Corollary 6.1.5 For a classical semiprime submodule, the following state- ments are equivalent: 1. completely semiprime, 2. symmetric, 3. IFP, 4. semi-symmetric.
21. is polynomially extensible if γ(M ) = M , then γ(M [x])
22. Theorem 7.2.1 If β(M ), N (M ), β co (M ), β s (M ) and β sc (M ) respectively denote the prime radical, s-prime radical, completely prime radical, strongly prime radical and strictly prime radical of M , then 1. β(M [x]) = β(M )[x],
23. N (M [x])
24. = N (M )[x], 3. β co (M [x]) = β co (M )[x],
25. β s (M [x]) = β s (M )[x] and 5. β sc (M [x]) ⊆ β sc (M )[x].
26. Proof: This is a simple application of Theorem 7.1.1 and the fact that the prime (resp. s-prime, completely prime, strongly prime and strictly prime) radical of a module is the intersection of all the prime (resp. s-prime, com- pletely prime, strongly prime and strictly prime) submodules. Proposition 7.2.1 Let γ be a module radical map such that γ(M [x])
27. ∩ M )[x] = γ(M [x]), i.e., γ satisfies the Amitsur property;
28. if γ(M ) = M , then γ(M [x])
29. = M [x], i.e., γ is polynomially extensible. Proof: (γ(M [x])
30. ∩M )[x] = (γ(M )[x]∩M )[x] = γ(M )[x] = γ(M [x]). Suppose γ(M ) = M , then by hypothesis, γ(M [
31. = γ(M )[x] = M [x].
32. Theorem 7.2.2 Let β, N , β co and β s denote respectively the prime, s-prime, completely prime and strongly prime module radicals. Then, β (resp. N , β co and β s ) satisfies the Amitsur property and is polynomially extensible.
33. ∩ R)[x] = γ(R[x]), i.e., γ satisfies the Amitsur property;
34. = R[x], i.e., γ is polynomially extensible. Theorem 7.2.4 The module R M is 2-primal (resp. β s -primal) if and only if so is R[x] M [x].
35. Proof: We prove the 2-primal case, the other case can proved in a similar way. If β(M ) = β co (M ), from Theorem 7.2.1, β(M [x])
36. = β(M )[x] = β co (M )[x] = β co (M [x]). For the converse, suppose β(M [x]) = β co (M [x]). Then β co (M ) = M ∩ β co (M )[x] = M ∩ β(M )[x] = β(M ). A comparison of Figure 8.2 of module radicals and that of ring radicals in [81] indicates lots of gaps in the figure for module radicals. This compels us to suggest that, more should be done in terms of investigating module analogues for existing ring radicals -where this is not possible, an explicit declaration of its nonexistence should be made. Theorem 8.1.1 If R M is a module, then 1. β cl = β = L = U = Rad whenever R is left Artinian;
37. R is both Artinian and commutative.
38. Proof: 1. From Figure 8.2, β cl ⊆ β ⊆ L ⊆ U ⊆ Rad. If R is left Artinian, by [9, Theorem 2.15] Rad ⊆ β cl from which the desired result follows.
39. For commutative rings, there is no distinction between: (a) completely prime, strictly prime, strongly prime, prime, l-prime and s-prime submodules;
40. classical prime and classical completely prime submodules.
41. M IFP or M symmetric,
42. R M cyclic (finitely generated, or free) with R medial (LSD, left per- mutable or right permutable).
43. Agayev N, Halicioglu S and Harmanci A. On reduced modules. Commun. Fac. Sci. Univ. Ank. Series. 58 (2009), 9-16.
44. Anderson D and Smith. E. Weakly prime ideals. Houston J. Math., 29 (2003), 831-840.
45. Andrunakievich V. A and Rjabuhin Ju. M. Special modules and special radicals, Soviet Math. Dokl, 3 (1962), 1790-1793. Russian original:Dokl. Akad. Nauk SSSR. 147 (1962), 1274-1277.
46. Argac N and Groenewald N. J. A generalization of 2-primal near-rings. Quaest. Math., 27 (2004), 397-413.
47. Atiyah, M. F and MacDonald I. G. Introduction to Commutative Algebra. New York: Addison-Wesley Publising Company, 1969.
48. Azizi A. Weakly prime submodules and prime submodules. Glasgow Math. J., 48 (2006), 343-346.
49. Azizi A. On prime and weakly prime submodules. Vietnam J. Math., 36 (2008), 315-325.
50. Beachy J. A. Some aspects of Noncommutative localization, Noncommu- tative Ring Theory, Kent State, 1975, Lecture Notes in Math. Springer- Verlag 545 (1976), 2-31.
51. Behboodi M. On weakly prime radical of modules and semi-compatible modules. Acta Math. Hungar., 113 (2006), 239-250.
52. Behboodi M. A generalization of Baer's lower nilradical for modules. J. Algebra Appl., 6 (2007), 337-353.
53. Behboodi M. On the prime radical and Baer's lower nilradical of modules. Acta Math. Hungar., 122 (2008), 293-306.
54. Behboodi M, Karamzadeh O. A. S and Koohy H. Modules whose certain ideals are prime. Vietnam J. Math., 32 (2004), 185-195.
55. Behboodi M and Koohy H. Weakly prime modules. Vietnam J. Math., 32 (2004), 303-317.
56. Behboodi M and Shojaee H. On chains of classical prime submodules. Bull. Iranian Math. Soc., 36 (2010), 149-166.
57. Beidar K. I, Fong. Y and Puczylowski E. R. On essential extensions of reduced rings and domains. Arch. Math., 83, no.4, (2004), 344-352.
58. Beidar K. I, Puczylowski E. R and Wiegandt R. Radicals and polynomial rings. J. Austral. Math. Soc., 72 (2002), 21-31.
59. Bell H. E. Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc., 2 (1970), 363-368.
60. Berrick A. J and Keating M. E. An introduction to rings and modules. Cambridge University Press, 2000.
61. Bican L, Kepka T and Nemec P. Rings, modules and preradicals. Lecture notes in pure and applied mathematics no.75, Marcel Dekker Inc., New York, 1982.
62. Birkenmeier G and Heatherly H. Medial rings and an associated radical. Czechoslovak Math. J., 40 (155) (1990), 258-283.
63. Birkenmeier G. F, Heatherly H. E and Lee E. K. Completely prime ide- als and associated radicals, in: S.K. Jain, S.T. Rizvi (Eds), Proceedings of the Biennial Ohio State-Denison Conference, 1992, World Scientific, Singapore, (1993), 102-129.
64. Blair W. D and Tsutsui H. Fully prime rings. Comm. Algebra, 22 (1994), 5389-5400.
65. Bourbaki N. Commutative algebra. Paris: Hermann, Publishers in Arts and Science, 1972.
66. Buys A. Uniformly strongly prime radicals. Quaest. Math., 21 1-2, (1998), 149-156.
67. Cohn P. M. Reversible rings. Bull. London Math. Soc., 31 (1999) 641-648.
68. Courter R. C. Rings all of whose factor rings are semiprime. Canad. Math. Bull., 12 (1969), 417-426.
69. Courter R. C. Fully idempotent rings have regular centroids, Proc. Amer. Math. Soc., 43 (2), (1974), 293-296.
70. Dauns J. Prime modules. Reine Angew. Math., 298 (1978), 156-181.
71. De La Rosa B and Veldsman S. A relationship between ring radicals and module radicals. Quaest. Math. 17 (1994), 453-467.
72. Ebrahimi S. A and Farzalipour F. On weakly prime submodules. Tamkang J. Math., 38 (2007), 247-252.
73. Ferrero M and Wisbauer R. Unitary strongly prime rings and related radicals. J. Pure Appl. Algebra. 181 (2003), 209-226.
74. France-Jackson H. On supernilpotent radicals with the Amitsur property. Bull. Aust. Math. Soc, 80 (2009), 423-429.
75. Gardner B. J and Wiegandt R. Radical theory for associative Rings. New York: Marcel Dekker, 2004.
76. Ghoneim S and Kepka T. Left self distributive rings generated by idem- potents. Acta. Math. Hungar., 101(1-2) (2003), 21-31.
77. Groenewald N. J. Contributions to the theory of group rings. PhD thesis, Rhodes University, 1978.
78. Groenewald N. J and Ssevviiri D. K .. othe's upper nil radical for modules.
79. Acta Math. Hungar., 138 (4), 2013, 295-306.
80. Groenewald N. J and Ssevviiri D. Completely prime submodules. Int. Electron. J. Algebra, 13, 2013, 1-14.
81. Groenewald N. J and Ssevviiri D. 2-primal modules, J. Algebra Appl., 12, 2013, DOI: 10.1142/S021949881250226X
82. Groenewald N. J and Ssevviiri D. Generalization of nilpotency of ring elements to module elements. Comm. Algebra, accepted.
83. Groenewald N. J and Ssevviiri D. Classical completely prime submodules, submitted.
84. Groenewald N. J and Ssevviiri D. The Amitsur property and polynomial equation for modules, submitted.
85. Handelman D and Lawrence J. Strongly prime rings. Trans. Amer. Math. Soc., 211 (1975), 209-223.
86. Hongan M. On strongly prime modules and related topics. Math. J. Okayama Univ., 24 (1982), 117-132.
87. Hua-Ping Y. On Quasi-duo rings. Glasgow Math. J., 37 (1995), 21-31.
88. Hwang U, Jeon Y. C and Lee Y. Structure and topological conditions of NI rings. J. Algebra, 302 (2006), 186-199.
89. Jenkins J and Smith P. F. On the prime radical of a module over a com- mutative ring. Comm. Algebra, 20 (1992), 3593-3602.
90. Juglal S and Groenewald N. J. Strongly prime near-ring modules. Arab. J. Sci. Eng., 36 (2011), 985-995.
91. Juglal S, Groenewald N. J and Lee K. S. E. Different prime R-ideals. Algebra Colloq., 17 (2010), 887-904.
92. Khuri S. M. Nonsingular retractable modules and their endomorphism rings. Bull. Austral. Math. Soc., 43 (1991), 63-71.
93. Kim N. K and Lee Y. Extensions of reversible rings. J. Pure Appl. Algebra, 185 (2003), 207-223.
94. Lambek J. On the presentation of modules by sheaves of factor modules. Canad, Math. Bull., 14 (1971), 359-368.
95. Lee T. K and Zhou Y. Reduced modules, Rings, Modules, Algebra and Abelian group. Lectures in Pure and Appl Math., 236 365-377, Marcel Decker, New York, 2004.
96. Loi N. V and Wiegandt R. On the Amitsur property of radicals, Algebra Discrete Math., 3 (2006), 92-100.
97. Marks G. On 2-primal ore extensions. Comm. Algebra, 29 (2001), 2113- 2123.
98. Marks G. Reversible and symmetric rings. J. Pure Appl. Algebra, 174 (2002), 311-318.
99. Marks G. A taxonomy of 2-primal rings. J. Algebra, 266 (2003), 494-520.
100. Matsumura H. Commutative ring theory. Cambridge: Cambridge Univer- sity Press, 1980.
101. McCasland R. L, Moore M. E and Smith P. F. On the spectrum of a module over a commutative ring. Comm. Algebra, 25 (1997), 79-103.
102. McCoy N. H. Certain classes of ideals in polynomial rings. Canad. J. Math., 9 (1957), 352-362.
103. McCoy N. H. The theory of rings. New York: Macmillan. 1964.
104. Meldrum J. D. P. Near-rings and their links with groups. Boston, London, Melbourne, Pitman, 1985.
105. Moghaderi J and Nekooei R. Strongly prime submodules and pseudo- valuation modules. Int. Electron. J. Algebra, 10 (2011), 65-75.
106. Naphipour A. R. Strongly prime submodules. Comm. Algebra, 37 (2009), 2193-2199.
107. Nicholson W. K and Watters J. F. The strongly prime radical. Proc. Amer. Math. Soc., 76 (1979), 235-240.
108. Nikseresht A and Azizi A. On radical formula in modules. Glasgow math. J., 53 (2011), 657-668.
109. Olson D. M and Lidi R. A uniformly strongly prime radical. J. Austral Math. Soc., 43 (1987), 95-102.
110. Pilz G. Near-rings. North-Holland, Amsterdam, New York, Oxford, 1983.
111. Reddy Y. V and Murty C. V. L. N. Semi-symmetric ideals in near-rings. Indian J. Pure Appl. Math., 16 (1985), 17-21.
112. Rege M. B and Buhphang A. M. On reduced modules and rings. Int. Elect. J. Algebra, 3 (2008), 58-74.
113. Rowen L. H. Ring theory. Academic Press, Inc. (San Diego, 1991).
114. Serre J. P. A course in Arithmetic. Springer-Verlag New York Inc., 1973.
115. Shin G. Prime ideals and sheaf representation of a pseudo symmetric ring. Trans. Amer. Math. Soc., 184 (1973), 43-60.
116. Ssevviiri D. On prime modules and radicals of modules. MSc. Treatise, Nelson Mandela Metropolitan University, 2011.
117. Ssevviiri D. Structure of non-nilpotent elements of some Z-modules. Int. J. Algebra, 6 (14) (2012), 691-695.
118. Stenstrom B. Rings of quotients, Springer-Verlag, Berlin, 1975.
119. Tuganbaev A. A. Multiplication modules over non-commutative rings. Sbornik: Mathematics, 194 (2003), 1837-864.
120. Tumurbat S and Wisbauer R. Radicals with the α-Amitsur property. J. Algebra Appl., 7 (2008), 347-361.
121. Van der walt A. P. J. Contributions to ideal theory in general rings. Proc. Kon. Ned. Akad. Wetensch., Ser. A., 67 (1964), 68-77.
122. Veldsman S. The superprime radical. In: Proc. Krems Conf. 1985, Con- tributions to General Algebra 4, 181-188, Wien & Stuttgart 1987.
123. Veldsman S. On a characterization of overnilpotent radical classes of near- rings by N-group. South African J. Sci., 89 (1991), 215-216.
124. Wiegandt R. Rings distinctive in radical theory. Quaest. Math., 22 (1999), 303-328.