On the Classical Zariski Topology Over Prime Spectrum of a Module
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Let R be an associative ring with identity and Spec(M) denote the set of all prime submodules of a right R-module M. In this paper, we study the classical Zariski topology on Spec(M) which is denoted by Τc. We prove that if (Spec(M), Τc) is a Noetherian topological space, then M has nitely many minimal prime submodules. We characterize all the irreducible components of (Spec(M), Τc) and all the minimal prime submodules of M for a non-zero at module M over a commutative ring R. We obtain some results concerning compactness and connectedness of (Spec(M), Τc) by using algebraic properties of the module M. We give some equivalent conditions for (Spec(M), Τc) to be a Hausdor¤ space or T1-space when M is a right module over a left perfect ring R.
Abstract
Let R be an associative ring with identity and Spec(M) denote the set of all prime submodules of a right R-module M. In this paper, we study the classical Zariski topology on Spec(M) which is denoted by Τc. We prove that if (Spec(M), Τc) is a Noetherian topological space, then M has nitely many minimal prime submodules. We characterize all the irreducible components of (Spec(M), Τc) and all the minimal prime submodules of M for a non-zero at module M over a commutative ring R. We obtain some results concerning compactness and connectedness of (Spec(M), Τc) by using algebraic properties of the module M. We give some equivalent conditions for (Spec(M), Τc) to be a Hausdor¤ space or T1-space when M is a right module over a left perfect ring R.
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