In this paper, we examine the algebraic properties of localizations  of commutative rings and how localizations affect the zero-divisor graphs  structure of modular rings. We also classify the zero-divisor graphs of modular  rings with respect to both the diameter and girth of their resultant zero-divisor  graphs

Cuchta, Thomas

Lokken, Kathryn A.

Young, William

English

Rose-Hulman Institute of Technology: Rose-Hulman Scholar

Rose-Hulman Undergraduate Mathematics Journal Volume 9 Issue 2 Article 3 Zero-divisor Graphs of Localizations and Modular Rings Thomas Cuchta Marshall University Kathryn A. Lokken, cuchta@marshall.edu Kathryn A. Lokken University of Wisconsin, Madison William Young Purdue University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Cuchta, Thomas; Lokken, Kathryn A.; and Young, William (2008) "Zero-divisor Graphs of Localizations and Modular Rings," Rose-Hulman Undergraduate Mathematics Journal: Vol. 9 : Iss. 2 , Article 3. Available at: https://scholar.rose-hulman.edu/rhumj/vol9/iss2/3 ZERO-DIVISOR GRAPHS OF LOCALIZATIONS AND MODULARRINGSTHOMAS CUCHTA, KATHRYN A. LOKKEN, WILLIAM YOUNGAbstract. In this paper, we examine the algebraic properties of localiza-tions of commutative rings and how localizations affect the zero-divisor graph’sstructure of modular rings. We also classify the zero-divisor graphs of modularrings with respect to both the diameter and girth of their resultant zero-divisorgraphs.1. IntroductionIn [4], Beck introduced the concept of the graph of a ring. Beck’s main purposewas to examine the colorings of commutative rings. The work was continued by D.D.Anderson and Naseer in [1]. In these two papers all ring elements were included inthe graph. However, this paper only includes the non-zero zero-divisors as verticesin the graph, just as D.F. Anderson and Livingston introduced in [2]. The diameterand girth of these zero-divisor graphs, among other things, were examined by D.F.Anderson and Livingston in [2], by Mulay in [9], and by DeMeyer and Schneiderin [6]. In [2] it was shown that all zero-divisor graphs of commutative rings mustbe connected with diameter less than or equal to three and girth three, four, orinfinity.Throughout, R will denote a commutative ring with unity. The powerset of Rwill be denoted P(R). In this paper we will only consider proper ideals of R. Aprime ideal P of a commutative ring R is an ideal of R such that if ab ∈ P, thena ∈ P or b ∈ P . A zero-divisor is an element z ∈ R such that zr = 0 for somenonzero r ∈ R. The set of zero-divisors of R will be denoted by Z(R), and Z(R)∗= Z(R)\{0}.Let R be a commutative ring with unity. Let S be a multiplicatively closedsubset of R. Define a binary relation ∼ on R × S by (r, s) ∼ (r′, s′) if and only ifthere exists s∗ ∈ R\S such that s∗(rs′ − r′s) = 0. This relation is an equivalencerelation. Define rs = (r, s). The localization of R at S is the set RS = {[ rs ] | r ∈ Rand s ∈ S} together with two binary operations +, · : RS × RS → RS defined by[ rs ] + [r′s′ ] = [rs′+r′sss′ ] and [rs ] · [ r′s′ ] = [rr′ss′ ]. By the usual convention, · is replaced byjuxtaposition, that is, x ·y = xy. Note that RS is a commutative ring with additiveidentity [01 ] and unity [11 ]. Notice that the complement of a prime ideal is alwaysmultiplicatively closed. By convention, a localization RR\P , where P is a primeideal, is denoted RP .The zero-divisor graph of R is denoted Γ(R) = {V,E}, where the vertices V ={z | z ∈ Z(R)∗} and the edges E = {(z, z′) | zz′ = 0 and z, z′ ∈ V } (edgesare sometimes denoted as (a, b) where a and b are vertices). A path in a graphis a sequence of vertices, a0, a1, ..., an such that each adjacent pair, ai, ai+1, is aDate: 18 August 2008.12 THOMAS CUCHTA, KATHRYN A. LOKKEN, WILLIAM YOUNGnonrepeated valid edge in the edge set; (an, a0) may or may not be an edge. Thedistance between two vertices in a graph is a path with the least number of edges.The diameter of a graph G, denoted diam(G), is the largest distance between anytwo vertices. A cycle is a path a0, a1, ..., an such that (an, a0) is an edge. The girthof a graph G, denoted g(G), is the length of a cycle with the least number of edges.A graph has girth ∞ if it contains no cycles. A graph is connected if there exists apath between all vertices of the graph.A reference for localizations can be found in [8], and a reference for graph theorycan be found in [5]. We compare the zero-divisor graphs of R and RP , the local-ization of R around P . We consider the localizations of modular rings, and showthey are isomorphic to a particular Zm. We also completely classify the zero-divisorgraphs of modular rings by diameter and girth.2. Γ(R) and Γ(RP )From the definition of ∼, an element [ rs ] of RS is in the equivalence class of [01 ]if and only if there exists an s∗ ∈ S such that s∗r = 0. Clearly, if [ rs ] = [01 ], thenfor any s′ ∈ S [ rs′ ] = [ 01 ].Lemma 1. If [ rs ] ∈ Z (RP ), then for any s¯ ∈ R\P , [ rs¯ ] ∈ Z (RP ).Proof. Assume [ rs ] ∈ Z(RP ). Then there exists [ r′s′ ] '= [01 ] such that [ rs ][ r′s′ ] = [01 ].So there exists s∗ ∈ R\P such that s∗rr′ = 0. Let [ rs¯ ] ∈ RP . Then since s∗rr′ = 0,[ rs¯ ][r′s′ ] = [01 ]. Thus, [rs¯ ] ∈ Z (RP ). !Define the numerator function n : RS → P(R) by n([ rs ]) = {r′ ∈ R | [ rs ] = [ r′s′ ]}.Lemma 2. If [ rs ] ∈ Z(RP ), then for every rˆ ∈ n([ rs ]), rˆ ∈ P ∩ Z(R).Proof. Suppose [ rs ] ∈ Z(RP ). Let [ r′s′ ] '= [01 ] such that [ rs ][ r′s′ ] = [01 ]. By definition,there exists s∗ ∈ R\P such that s∗rr′ = 0, so r ∈ Z(R). We know for everyrˆ ∈ n([ rs ]), [ rs ] = [ rˆsˆ ] for some sˆ. So, suppose rˆ '∈ P . Then we know that s∗rˆ '∈ P ,since P is a prime ideal. If s¯ = s∗rˆ, then s¯r′ = 0, which implies that [r′s′ ] = [01 ], acontradiction. Thus rˆ ∈ P . !Definition 3. The total quotient ring of R, denoted T (R), is the localization RSwhere S = R\Z(R).The following lemmas present some relations between elements in T (R) andelements in RP .Lemma 4. Assume Z (R) ⊆ P . Then the following are equivalent.i) [ r1 ][r′1 ] = [01 ] in T (R).ii) [ rs ][r′s′ ] = [01 ] in T (R) for all s, s′ ∈ R\Z (R).iii) [ rs ][r′s′ ] = [01 ] in RP for all s, s′ ∈ R\P .iv) rr′ = 0.Proof. (i⇒ ii) Assume [ r1 ][ r′1 ] = [01 ] in T (R) . Then there exists s∗ ∈ R\Z (R) suchthat s∗rr′ = 0. But s∗ '∈ Z(R), so rr′ = 0 which implies [ rs ][ r′s′ ] = [01 ] in T (R) forany s, s′ ∈ R\Z(R).(ii⇒ iii) Assume [ rs ][ r′s′ ] = [01 ] in T (R) for all s, s′ ∈ R\Z (R) . Then there existss∗ ∈ R\Z (R) such that s∗rr′ = 0. Since s∗ /∈ Z (R) , we know rr′ = 0. Therefore,[ rs ][r′s′ ] = [01 ] in RP for all s, s′ ∈ R\P .ZERO-DIVISOR GRAPHS OF LOCALIZATIONS AND MODULAR RINGS 3(iii ⇒ iv) Assume [ rs ][ r′s′ ] = [01 ] in RP . Then there exists s∗ ∈ R\P such thats∗rr′ = 0. We know s∗ '∈ Z(R) since Z(R) ⊆ P , so rr′ = 0.(iv ⇒ i) Clear. !The following lemma shows that a pair of equivalence classes that are equal inRP are also equal in T (R), under a particular condition.Lemma 5. Assume Z (R) ⊆ P . Then [ rs ] = [ r′s′ ] in RP if and only if [rs ] = [r′s′ ] inT (R).Proof. (⇒) Assume [ rs ] = [ r′s′ ] in RP . Then there exists s∗ ∈ R\P such thats∗(rs′ − r′s) = 0. Since s∗ /∈ Z (R) , we know rs′ − r′s = 0. Hence, [ rs ] = [ r′s′ ] inT (R).(⇐) Assume [ rs ] = [ r′s′ ] in T (R). Then there exists s∗ ∈ R\Z (R) such thats∗(rs′ − r′s) = 0. Since s∗ /∈ Z (R) , we know rs′ − r′s = 0. Thus, [ rs ] = [ r′s′ ] inRP . !The next lemma establishes an expected result between zero-divisors of R andtheir respective canonical fractions.Lemma 6. Assume Z (R) ⊆ P . Let r, r′ ∈ Z (R)∗ . Then, r = r′ if and only if[ r1 ] = [r′1 ] in RP .Proof. (⇐) Assume [ r1 ] = [ r′1 ] in RP . Then, there exists s∗ ∈ R\P such thats∗(r − r′) = 0. Since s∗ '∈ Z (R), we know r − r′ = 0, which implies r = r′.(⇒) Trivial. !It would be convenient to be able to create a homomorphism between R andT (R), and the next lemma establishes that under certain circumstances, the obviouscandidate for a homomorphism will suffice.Lemma 7. If R = Z(R) ∪ U(R), then for every [ rs ] ∈ T (R), there exists r′ ∈ Rsuch that [ rs ] = [r′1 ].Proof. Consider r′ = s−1r. Then, for all s∗ ∈ R\Z(R), we have s∗(r − sr′) =s∗(r − ss−1r) = 0, which implies [rs ] = [ r′1 ]. !A generalized version of the following theorem was presented and proved asTheorem 2.2 in [3]. However, we will now present a different proof below.Theorem 8. If R is a commutative ring with identity such that R = Z(R)∪U(R),then R ∼= T (R).Proof. Consider the function φ : R → T (R) defined by φ(r) = [ r1 ]. Consider[ r1 ], [r′1 ] ∈ T (R) such that [ r1 ] = [ r′1 ]. Then, there exists s∗ ∈ R\Z(R) such thats∗(r − r′) = 0. But s∗ '∈ Z(R), so r − r′ = 0 implies r = r′, thus φ is injective.Consider any [ rs ] ∈ T (R). By Lemma 7, we know that [ rs ] = [ r′1 ] for some r′ ∈ R.Thus, φ(r′) = [ r′1 ] = [rs ], so φ is surjective. Let r, r′ ∈ R. Then, φ(r+r′) = [ r+r′1 ] =[ r1 ]+[r′1 ] = φ(r)+φ(r′) and φ(rr′) = [ rr′1 ] = [r1 ][r′1 ] = φ(r)φ(r′), so φ is an operationpreserving function. Thus, R ∼= T (R). !Note that since R ∼= T (R), it follows trivially that Γ(R) ∼= Γ(T (R)) under theabove conditions.4 THOMAS CUCHTA, KATHRYN A. LOKKEN, WILLIAM YOUNG3. Localizations of ZnConsider Zn, where n=p1e1p2e2 ...pkek and each pi is prime. Since Zn is a princi-pal ideal ring, every ideal is generated by a single element. Notice that prime idealsin Zn will be generated by prime factors of n. Thus, we will only be concerned onlywith localizations of Zn around ideals of the form (pi), where 1 ≤ i ≤ k.Lemma 9. If [ qpiei+r1 ] ∈ Zn(pi), then [ qpiei+r1 ] = [r1 ].Proof. All elements in Zn can be written as qpiei + r for some 0 ≤ r < pieiby the division algorithm. Thus [qpiei+r1 ] = [r1 ], since if s∗ ∈ Zn\(pi) such thats∗ = p1e1 ...pi−1ei−1pi+1ei+1 ...pkek , then s∗qpiei = 0 in Zn. !The following lemma is in a similar spirit to Lemma 7. It establishes that wecan use the canonical mapping between Zn and prime ideal localizations of Zn asthe basis of an isomorphism.Lemma 10. If [ rs ] ∈ Zn(pi), then [ rs ] = [ r′1 ] in Zn(pi) for some r′ ∈ Zn.Proof. Let [ rs ] ∈ Zn(pi). Since s ∈ Zn\(pi), gcd(peii , s) = 1, and thus (piei) ∼= Zs.So, r ≡ mpiei(mod s) for some m. Therefore, there exists r′ = m′piei for somer′,m′ ∈ Zn such that sr′ ≡ r − mpeii (mod s). Hence mpiei ≡ r − sr′(mod s).Then, there exists s¯ ∈ Zn\(pi), namely s¯ = p1e1 ...pi−1ei−1pi+1ei+1 ...pkek , such thats¯(r− sr′) = s¯(mpeii − sm′peii ) = 0 in Zn which, by definition, implies [rs ] = [ r′1 ]. !In general, it is a difficult task to determine a ring that is isomorphic to agiven localization. The following theorem yields a very simple isomorphism forlocalizations around prime ideals of modular rings.Theorem 11. Let n=p1e1p2e2 ...pkek . Then Zn(pi) ∼= Zpiei .Proof. Consider φ : Zpiei → Zn(pi) defined by φ(r) = [ r1 ]. Assume φ(r) = φ(r¯).Then [ r1 ] = [r¯1 ]. By definition, there exists an s∗ ∈ Zn\(pi) such that s∗(r− r¯) = 0.Thus, peii must divide r − r¯, which is impossible unless r = r¯. So, φ is injective.Let [ rs ] ∈ Zn(pi). Then by the previous lemma, we know [rs ] = [ r′1 ] for somer′ ∈ Zn. So, r′ ≡ m (mod n). Then, since n = pe11 . . . pekk , r′ ≡ m (mod piei).Then, φ(r′) = φ(m) = [m1 ], and since r′ ≡ m (mod n), this implies [m1 ] = [ r′1 ] = [rs ],and hence φ is surjective.Take r, r¯ ∈ Zpiei . Then, φ(r + r¯) = [ r+r¯1 ] = [ r1 ] + [ r¯1 ] = φ(r) + φ(r¯). Similarly,φ(rr¯) = [ rr¯1 ] = [r1 ][r¯1 ] = φ(r)φ(r¯). !Corollary 12. Let n = p1e1p2e2 ...pkek . Then Γ(Zn(pi)) ∼= Γ(Zpiei ).The isomorphism established above inspires a question about whether localiza-tions around non-prime ideals have other nice isomorphisms. It also hints at thepossibility of an unexplored generalization of the theorem to localizations of directproduct decompositions, since a modular ring can be decomposed into a directproduct of powers of primes.ZERO-DIVISOR GRAPHS OF LOCALIZATIONS AND MODULAR RINGS 54. Classification of Γ(Zn)Theorem 13. The following table holds true:Factorization of n Diameter Girthp; p is prime - -22 0 ∞32 1 ∞p2; p is prime and p > 3 1 323, or 2p; p odd prime 2 ∞pq; p,q, distinct odd primes 2 4pm; p is prime, m > 2, and pm '= 8 2 34p; p is an odd prime 3 4pqk; p, q distinct primes, k ∈ Z+ and pqk does 3 3not meet any criteria listed aboveProof. Let n = p where p is prime, then Γ(Zn) = ∅, since Zp is a field.Let n = 22, then diam(Γ(Zn)) = 0 and g(Γ(Zn)) =∞ by observation.Let n = 32, then diam(Γ(Zn)) = 1 and g(Γ(Zn)) =∞ by observation.Let n = p2 where p is prime and p > 3. Then all zero-divisors of Zp2 are multiplesof p. Consider the zero-divisors m1p, m2p, and m3p. Then, there is a 3-cycle m1p— m2p — m3p — m1p and it is clear that all zero-divisors are attached to eachother, so diam(Γ(Zn)) = 1 and g(Γ(Zn)) = 3.Let n = 23, then diam(Γ(Zn)) = 2 and g(Γ(Zn)) =∞ by observation.Let n = 2p where p is an odd prime. Thus, Zn ∼= Z2 × Zp. By Theorem 2.5 in[2], we know that Zn is a star graph. Thus diam(Γ(Zn)) = 2 and g(Γ(Zn)) =∞.Let n = pq where p and q are distinct odd primes. Then clearly, Γ(Zn) is completebipartite since Zp and Zq are fields, so diam(Γ(Zn)) = 2 and g(Γ(Zn)) = 4.Let n = pm where p is prime, m > 2, and pm '= 8. Then, since multiples of p arezero-divisors, consider m1p and m2p, any two arbitrary multiples of p. There willalways be a 2-path m1p — pm−1 — m2p and a 3-cycle m1pm−1 — p — m2pm−1— m1pm−1. Thus diam(Γ(Zn)) = 2 and g(Γ(Zn)) = 3.Let n = 4p. Then Z(Zn) = {p!, 2p!, 3p!, 2 ·2, 2 ·3, . . . , 2 · (p−1), 2 · (p+1), . . . , 2 ·(n − 1)} for all " ∈ N. Consider 2, 2p!1 , p!2m1, and 2m2 where m1 ∈ {1, 3}, m2is an even element, and "i ∈ N. There is a shortest 3-path, namely p!1m1 — 2m2— 2p!2 — 2, so diam(Γ(Zn)) = 3. Now consider 2m3 where m3 is a distinct evenelement such that m3 '= m2. There is a 4-cycle p!1m1 — 2m2 — 2p!2 — 2m3— p!1m1. Since the multiples of p must connect to some multiple of 2, and allmultiples of 2 must connect to a multiple of p, any cycle in Γ(Zn) must have aneven number of edges. So, since we have a exhibited a 4-cycle and the girth cannotbe 3, g(Γ(Zn)) = 4.Let n = pqk; p, q distinct primes, k ∈ Z+ and pqk does not meet any criterialisted above. Then, p — qk — pk — q is a shortest 3-path, and qk — pq — pk isa 3-cycle. Thus diam(Γ(Zn)) = 3 and g(Γ(Zn)) = 3. !5. Conclusions and AcknowledgmentsLocalizations provide a valuable way to extend rings in general commutative ringtheory. Looking at the implications to zero-divisor graphs may provide a deeperunderstanding of the structure of zero-divisor graphs and, in turn, of the zero-divisors themselves. The classification of modular rings is a step that we hope will6 THOMAS CUCHTA, KATHRYN A. LOKKEN, WILLIAM YOUNGlead to a more complete classification of zero-divisor graphs, relying less on numbertheoric devices and more on broad algebraic properties.This paper was written over two summers (2007-2008) at the REU program atWabash College. Both programs were supported by NSF grants DMS-0453387 andDMS-0755260. The authors would like to thank Michael Axtell and Joe Sticklesfor being advisors over this paper and aiding in understanding of material.References[1] Anderson, D.D. and Naseer, M., Beck’s coloring of a commutative ring, J. Algebra 159 (1993),500-514.[2] Anderson, David F. and Livingston, Philip S., The zero-divisor graph of a commutative ring,J. Algebra 217 (1999), 434-447.[3] Anderson, David F., Levy, Ron, and Shapiro, Jay, Zero-divisor graphs, von Neumann regularrings, and Boolean algebras, Journal of Pure and Applied Algebra 180(3) (2003), 221-241.[4] Beck, I., Coloring of commutative rings, J. Algebra 116 (1988), 208-226.[5] R. Diestel, Graph Theory, Springer-Verlag, 1997.[6] DeMeyer, F.and Schneider, K., Automorphisms and zero-divisor graphs of commutative rings,International J. of Commutative Rings 1(3) (2002), 93-106.[7] Dummit, David S., and Foote, Richard M., Abstract Algebra, 3rd Ed. (2004).[8] Kaplansky, Irving, Commutative Rings, Univ. of Chicago Press, Chicago, rev. ed., 1974.[9] Mulay, S.B., Cycles and symmetries of zero-divisors, Comm. Alg. 30(7) (2002), 3533-3558.

Zero-divisor Graphs of Localizations and Modular Rings

https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1164&amp;context=rhumj

Zero-divisor Graphs of Localizations and Modular Rings

Abstract

Similar works

Full text

Available Versions

Rose-Hulman Institute of Technology: Rose-Hulman Scholar