Distributive and multiplication modules. (English) Zbl 1228.13015
In a paper by A. Barnard [J. Algebra 71, 174–178 (1981; Zbl 0468.13011)], \(M\) is called a multiplication module over a commutative ring \(R\) with identity means here that every submodule has the form \(IM\) for some ideal \(I\) of \(R\). An \(R\)-module \(M\) is said to be a distributive module if for all submodules \(N\), \(K\), and \(L\) of \(M\) we have, \(N+(K\cap L)=(N+K)\cap (N+L)\).
In the paper under review the authors show that every distributive module satisfies the radical formula. Also they provide a proof of the prime avoidance theorem for distributive modules. In addition, they study the question when a multiplication module becomes flat.
In the paper under review the authors show that every distributive module satisfies the radical formula. Also they provide a proof of the prime avoidance theorem for distributive modules. In addition, they study the question when a multiplication module becomes flat.
Reviewer: Siamak Yassemi (Tehran)
MSC:
13C10 | Projective and free modules and ideals in commutative rings |
13C11 | Injective and flat modules and ideals in commutative rings |
13C13 | Other special types of modules and ideals in commutative rings |
Citations:
Zbl 0468.13011References:
[1] | Abu-Saymeh S.: On dimensions of finitely generated modules. Comm. Algebra 23(3), 1131–1144 (1995) · Zbl 0833.13004 · doi:10.1080/00927879508825270 |
[2] | Ali M.M., Smith D.J.: Projective, flat and multiplication modules. New Zealand J. Math 31(2), 115–129 (2002) · Zbl 1085.13004 |
[3] | Anderson D.D., Al-Shaniafi Y.: Multiplication modules and the ideal {\(\theta\)}. Comm. Algebra 30(7), 3383–3390 (2002) · Zbl 1016.13002 · doi:10.1081/AGB-120004493 |
[4] | Auslander M., Goldman O.: Maximal orders. Trans. Amer. Math. Soc 97, 1–24 (1960) · Zbl 0117.02506 · doi:10.1090/S0002-9947-1960-0117252-7 |
[5] | Azizi, A.: Weak multiplication modules. Czechoslovak Math. J. 53(128)(3), 529–534 (2003) · Zbl 1083.13502 |
[6] | Barnard A.: Multiplication modules. J. Algebra 71(1), 174–178 (1981) · Zbl 0468.13011 · doi:10.1016/0021-8693(81)90112-5 |
[7] | El-Bast Z.A., Smith P.F.: Multiplication modules. Comm. Algebra 16(4), 755–779 (1988) · Zbl 0642.13002 · doi:10.1080/00927878808823601 |
[8] | Gaur A., Maloo A.K., Parkash A.: Prime submodules in multiplication modules. Int. J. Algebra 1(5-8), 375–380 (2007) · Zbl 1121.13012 |
[9] | Lu C.P.: Unions of prime submodules. Houston J. Math 23(2), 203–213 (1997) · Zbl 0885.13004 |
[10] | Matsumura H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1989) · Zbl 0666.13002 |
[11] | Naoum A.G.: Flat modules and multiplication modules. Period. Math. Hungar 21(4), 309–317 (1990) · Zbl 0739.13005 · doi:10.1007/BF02352695 |
[12] | Sharif H., Sharifi Y., Namazi S.: Rings satisfying the radical formula. Acta Math. Hungar 71(1–2), 103–108 (1996) · Zbl 0890.13001 · doi:10.1007/BF00052198 |
[13] | Smith W.W.: Projective ideals of finite type. Canad. J. Math 21, 1057–1061 (1969) · Zbl 0183.04001 · doi:10.4153/CJM-1969-116-7 |
[14] | Stephenson W.: Modules whose lattice of submodules is distributive. Proc. London Math. Soc (3) 28, 291–310 (1974) · Zbl 0294.16003 · doi:10.1112/plms/s3-28.2.291 |
[15] | Vasconcelos W.V.: On projective modules of finite rank. Proc. Amer. Math. Soc 22(2), 430–433 (1969) · Zbl 0176.31601 · doi:10.1090/S0002-9939-1969-0242807-2 |
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