Prime submodules of Noetherian modules. (English) Zbl 0814.16017

Let \(M\) be a left module over a ring \(R\). Then a proper submodule \(N\) of \(M\) is defined to be a prime submodule if for each \(r \in R\), \(m \in M\), \(rRm \subseteq N\) implies \(rM \subseteq N\) or \(m \in N\). Hence if \(N\) is a prime submodule of \(M\), then the annihilator \(P\) of \(M/N\) is a two- sided prime ideal of \(R\). A proper submodule \(K\) of \(M\) is called a strongly prime submodule of \(M\) if the annihilator \(Q\) of \(M/K\) satisfies the following conditions: (i) \(Q\) is a prime ideal of \(R\) and \(R/Q\) is left Goldie; (ii) \(M/K\) is a torsion-free left \((R/Q)\)-module. A strongly prime submodule of \(M\) is prime. The authors study the question, which prime ideals \(P\) (resp. \(Q\)) of \(R\) are the annihilators of \(M/N\) with \(N\) prime (strongly prime) in \(M\). They also obtain interesting results on chain conditions for (strongly) prime submodules, for example, it is shown that if a ring \(R\) has ACC (resp. DCC) on prime ideals then any finitely generated left \(R\)-module satisfies ACC (resp., DCC) on strongly prime submodules.


16P40 Noetherian rings and modules (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16D25 Ideals in associative algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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[1] F.W. Anderson and K.R. Fuller, Rings and categories of modules , Springer-Verlag, New York, 1974. · Zbl 0301.16001
[2] J. Dauns, Prime modules , J. Reine Angew. Math. 298 (1978), 156-181. · Zbl 0365.16002
[3] ——–, Prime modules and one-sided ideals , in Ring theory and algebra III, Proceedings of the Third Oklahoma Conference (B.R. McDonald, ed.), Dekker, New York, 1980, 301-344. · Zbl 0457.16021
[4] Z.A. El-Bast and P.F. Smith, Multiplication modules , Comm. Algebra 16 (1988), 755-779. · Zbl 0642.13002
[5] K.R. Goodearl and R.B. Warfield, An introduction to noncommutative Noetherian rings , London Math. Soc. Student Texts 16 , Cambridge University Press, Cambridge, 1989. · Zbl 0679.16001
[6] K. Koh, On one sided ideals of a prime type , Proc. Amer. Math. Soc. 28 (1971), 321-329. · Zbl 0209.07402
[7] ——–, On prime one-sided ideals , Canad. Math. Bull. 14 (1971), 259-260. · Zbl 0212.38104
[8] C.-P. Lu, Prime submodules of modules , Comm. Math. Univ. Sancti Pauli 33 (1984), 61-69. · Zbl 0575.13005
[9] ——–, \(M\)-radicals of submodules in modules , Math. Japon. 34 (1989), 211-219. · Zbl 0673.13007
[10] R.L. McCasland and M.E. Moore, On radicals of submodules , Comm. Algebra 19 (1991), 1327-1341. · Zbl 0745.13001
[11] ——–, Prime submodules , Comm. Algebra 20 (1992), 1803-1817. · Zbl 0776.13007
[12] J.C. McConnell and J.C. Robson, Noncommutative Noetherian rings , Wiley, Chichester, 1987. · Zbl 0644.16008
[13] G. Michler, Prime right ideals and right Noetherian rings , Proc. Symposium on Theory of Rings, 1971, in Ring theory (R. Gordon, ed.), Academic Press, New York, 1972, 251-255. · Zbl 0235.16027
[14] P.F. Smith, The injective test lemma in fully bounded rings , Comm. Algebra 9 (1981), 1701-1708. · Zbl 0464.16017
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