Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Volume 11, Number 4 (2016), pp. 417-421 © Research India Publications http://www.ripublication.com
Equivalent Conditions for Disconnectedness of Prime Spectrum of a Ring
A.V. S. N. Murty and M. N. Srinivas Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-632014, Tamil Nadu, India. E-mail: [email protected] , [email protected]
Abstract In this paper, it is proved that if R is a semi-prime ring, then a X() a is an isomorphism of ER(), the set of all idempotent elements of , onto Boolean algebra of clopen subsets of Spec . And it is studied about the equivalent conditions for Prime Spectrum of a ring is disconnected.
Keywords: Prime spectrum, semi-prime ring, clopen subsets, disconnected. AMS Mathematics Subject Classifications (2010): 06E15, 16D25.
1. INTRODUCTION Let be a commutative ring with identity and X be the set of all prime ideals of . For any AR , let XAPXAP() . It is clear that XAAR() forms a topology on , for which X() a a R is a base ([1], [2] and[ 5]). The set of all prime ideals of a commutative ring with identity together with this topology is called the prime spectrum of a ring and is denoted by Spec R ([4], [6] and [8]). A. V. S. N. Murty etc., all ([10] and [11]) studied the properties of Prime spectrum of a ring and some homeomorphic theorems on Prime spectrum of a ring. Also A. V. S. N. Murty etc., all [10] characterized certain ring theoretic properties of by the corresponding topological properties of Spec . For example, a semi prime ring is 418 A.V. S. N. Murty and M. N. Srinivas regular if and only if Spec R is a Hausdorff space. In this paper, it is proved that if is a semi-prime ring, then a X() a is an isomorphism of ER()onto Boolean algebra of clopen subsets of Spec . And it is studied about the equivalent conditions for prime spectrum of a ring is disconnected. Throughout this paper we consider only commutative rings with identity and hence we preferred to call these as rings for simplicity.
2. PRELIMINARIES Definition 2.1: An ideal P of a ring R is called prime if (1) PR (2) For any a, b R , ab Pimplies either aP or bP Definition 2.2: Let R be a ring, an ideal M of R is said to be maximal if (1) MR (2) For any ideal J of such that MJR implies either MJ or JR
Definition 2.5: Let be a ring. An element aR is said to be nilpotent if there exist a positive integer n such that an 0 .
Definition 2.6: An ideal I of a ring is said to be nil ideal of if each element of is nilpotent.
Theorem 2.7: The set N of all nilpotent elements of a ring is an ideal and RN/ has no nilpotent element other than zero element in . Also N coincides with the intersection of all prime ideals of . Here N is called the nil radical of and is denoted by NR().
Definition 2.8: A ring is called semi prime if NR( ) {0}. Theorem 2.9: Let be a ring. For any a,() b E R define a b a(1 b ) b (1 a ) . Then ER()is a Boolean ring with respect to ‘+’ as addition and multiplication is same as in .
Theorem 2.10: a a N() R is an isomorphism of onto ERNR( / ( )).
Equivalent Conditions for Disconnectedness of Prime Spectrum of a Ring 419
Theorem 2.11: Let R be a ring then the following are equivalent. 2 (1) There exists aR such that aa and a 0,1.
(2) RRR12, where RR12, are non-trivial rings.
Definition 2.12: A topological space X is said to be connected if the only subsets of which are both open and closed (clopen sets) are and the empty set . Otherwise it is called disconnected.
3. MAIN THEOREMS: Theorem 3.1: If R is semi-prime, then a X() a is an isomorphism of ER()onto Boolean algebra of clopen subsets ofSpec R .
Proof: Let R be a semi-prime ring; i.e., NR( ) {0}.Let BR(Spec ) be the set of all clopen subsets of Spec R . Define f: E ( R ) B (Spec R ) by f()() a X a , for all a E() R . We shall prove that f is an isomorphism. First to prove that is one-one.
Let a12,() a E R . Suppose, f()() a12 f a .
Then X()() a12 X a r()() a12 r a (by theorem2.6 (xvi) [9] )
a12 r() a and a21 r() a m n a12 r a and a21 r a for some m, n Z
aa12 and aa21
aa12 Therefore is one-one. Next to prove that is onto. Let BR(Spec ) . Write UXI ()and UXJ (), where I and J are ideals of . Then XIXJXIJ()()() and XXIXJXIJ()()() . Therefore RIJand hence, there exists a E() R such that Ia . Now, U X()() I X a X()() a f a . Therefore is onto. Finally to prove that is a homomorphism.
For any a12,() a E R , faa(.)1 2 Xaa (.) 1 2 Xa () 1 Xa () 2 fafa ().() 1 2
and faa()()()()()()1 2 Xaa 1 2 Xa 1 Xa 2 fa 1 fa 2 . Therefore is homomorphism. Thus is isomorphism. 420 A.V. S. N. Murty and M. N. Srinivas
Theorem 3.2: Let R be a ring then the following are equivalent. (1) There exists aR such that aa2 and a 0,1. (2) Spec is disconnected.
Proof: (1) (2) Suppose there exists aR such that and . Let XR Spec . Let Xa()be a basic open set in X . To prove that X is disconnected. For this we have to prove that Xa() , X() a X and Xa()is clopen in X . First to prove . Suppose Xa() . Then aP for all prime ideals P of R and hence, a N( R ) an 0 , for some nZ which implies that a 0 which is absurd since a 0 . Therefore, . Next to prove that . Suppose, X() a X . Then aP for all prime ideals of , which implies that a is a unit; i.e., ab 1for some b R a a2 b ab 1 which is absurd since a 1. Therefore, . Finally to prove that Xa() is open. For this, to prove that X(1 a ) X X ( a ) . Let PXXa( ) PXa ( ) aP 1 aPPX (1 a ). Therefore and hence is clopen in X . Thus Spec is disconnected. (2) (1) Suppose Spec is disconnected. Then Spec S is disconnected, where SRNR /() (since Spec(RNRR / ( )) Spec ). Therefore, there exists clopen set U in Spec such that U and US Spec and hence there exists u E() S such that X() u U and u 0,1(by theorem (3.1)). Therefore there exists a E() R such that a N() R u and a 0,1 (by theorem 2.10). Hence, there exists aR such that and . The following is the immediate consequences of the above Theorem 3.2 and the Theorem 2.11.
Theorem 3.3: Let be a ring. Then the following are equivalent.
(1) RRR12, where RR12, are non-trivial rings. (2) Spec is disconnected.
Future works: It can be studied the above properties for fuzzy prime spectrum of a ring also. Equivalent Conditions for Disconnectedness of Prime Spectrum of a Ring 421
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