PRIME BASES OF WEAKLY PRIME SUBMODULES AND THE WEAK RADICAL OF SUBMODULES Nikseresht, Ashkan; Azizi, Abdulrasool;
We will introduce and study the notion of prime bases for weakly prime submodules and utilize them to derive some formulas on the weak radical of submodules of a module. In particular, we will show that every one dimensional integral domain weakly satisfies the radical formula and state some necessary conditions on local integral domains which are semi-compatible or satisfy the radical formula and also on Noetherian rings which weakly satisfy the radical formula.
prime basis of a submodule;weakly prime submodule;prime submodule;radical formula;
A. Azizi, On prime and weakly prime submodules, Vietnam J. Math. 36 (2008), no. 3, 315-325.
A. Azizi, Radical formula and prime submodules, J. Algebra 307 (2007), no. 1, 454-460.
A. Azizi, Radical formula and weakly prime submodules, Glasg. Math. J. 51 (2009), no. 2, 405-412.
A. Azizi and A. Nikseresht, Simplified radical formula in modules, Houston J. Math. 38 (2012), no. 2, 333-344.
M. Behboodi, On weakly prime radical of modules and semi-compatible modules, Acta Math. Hungar. 113 (2006), no. 3, 243-254.
M. Behboodi and H. Koohi, Weakly prime modules, Vietnam J. Math. 32 (2004), no. 2, 185-195.
K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), no. 3, 285-293.
S. H. Man, On commutative Noetherian rings which satisfy the generalized radical for- mula, Comm. Algebra 27 (1999), no. 8, 4075-4088.
R. McCasland and M. Moore, On radicals of submodules of finitely generated modules, Canad. Math. Bull. 29 (1986), no. 1, 37-39.
A. Nikseresht and A. Azizi, On arithmetical rings and the radical formula, Vietnam J. Math. 38 (2010), no. 1, 55-62.
H. Sharif, Y. Sharifi, and S. Namazi, Rings satisfying the radical formula, Acta Math. Hungar. 71 (1996), no. 1-2, 103-108.
P. F. Smith, Primary modules over commutative rings, Glasg. Math. J. 43 (2001), no. 1, 103-111.