Abstract
Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).
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References
D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc., 122 (1994), 13–14.
Bernard, Multiplication modules, Journal of Algebra, 71 (1981), 174–178.
M. Behboodi and H. Koohy, On minimal prime submodules, Far East J. Math. Sci., 6 (2002), 83–88.
M. Behboodi, O. A. Karamzadeh and H. Koohy, Modules whose certain submodules are prime, Vietnam Journal of Mathematics, 32 (2004), 303–317.
M. Behboodi and H. Koohy, Weakly prime modules, Vietnam Journal of Mathematics, 32 (2004), 185–195.
J. Dauns, Prime modules, J. Reine Angew. Math., 298 (1978), 156–181.
Z. E. El-Bast and P. F. Smith, Multiplication modules, Comm. Alg., 16 (1988), 755–779.
E. H. Feller and E. W. Swokowski, Prime modules, Canad. J. Math., 17 (1965), 1041–1052.
K. R. Goodearl, An Introduction to Non-commutative Noetherian Rings, London Math. Soc. Student texts 16, Cambridge University Press (Cambridge, 1989).
J. Jenkins and P. F. Smith, On the prime radical of a module over a commutative ring, Comm. Alg., 20 (1992), 3593–3602.
I. Kaplansky, Commutative Rings, London Dillon’s Q.M.C. Bookshop (1917).
H. I. Karakas, On Noetherian modules, Metu. Journal of Pure and Applied Science, 5 (1972), 165–168.
C. P. Lu, Prime submodules of modules, Comm. Math. Univ. Sancti. Pauli., 33 (1995), 61–69.
R. L. McCasland and P. F. Smith, Prime submodules of Noetherian modules, Rocky Mountain J. Math., 23 (1993), 1041–1062.
W. K. Nicholson, Very semisimple modules, Amer. Math. Mon., 104 (1997), 159–162.
Y. Tiras and M. Alkan, Prime modules and submodules, Comm. Alg., 31 (2003), 5253–5261.
Y. Tiras, A. Harmanci and P. F. Smith, A characterization of prime submodules, Journal of Algebra, 212 (1999), 743–752.
R. Wisbauer, On prime modules and rings, Comm. Algebra, 11 (1983), 2249–2265.
W. V. Vasconcelos, Finiteness in projective ideals, J. Algebra, 25 (1973), 269–278.
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Koohy, H. On finiteness of multiplication modules. Acta Math Hung 118, 1–7 (2008). https://doi.org/10.1007/s10474-007-6136-0
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DOI: https://doi.org/10.1007/s10474-007-6136-0