14_628_behboodi.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 2, 2010, 303-316 ISSN 1307-5543 – www.ejpam.com On the Structure of Commutative Rings with p1 k1 · · · pn kn (1≤ ki ≤ 7) Zero-Divisors M. Behboodi1,2∗ and R. Beyranvand1 1 Department of Mathematical Science, Isfahan University of Technology, Isfahan, Iran 2 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran Abstract. Let R be a finite commutative ring with identity and Z(R) denote the set of all zero-divisors of R. Note that R is uniquely expressible as a direct sum of local rings Ri (1 ≤ i ≤ m) for some m ≥ 1. In this paper, we investigate the relationship between the prime factorizations |Z(R)| = p1 k1 · · · pn kn and the summands Ri . It is shown that for each i, |Z(Ri)| = p j t j for some 1 ≤ j ≤ n and 0 ≤ t j ≤ k j . In particular, rings R with |Z(R)| = pk where 1 ≤ k ≤ 7, are characterized. Moreover, the structure and classification up to isomorphism all commutative rings R with |Z(R)| = p1 k1 . . . pn kn , where n ∈ N, p , i s are distinct prime numbers, 1 ≤ ki ≤ 3 and nonlocal commutative rings R with |Z(R)| = pk where k = 4 or 5, are determined. 2000 Mathematics Subject Classifications: 16B99; 13A99; 68R10 Key Words and Phrases: Finite ring, Zero-divisor, Local rings 1. Introduction Throughout the paper R always denotes a commutative ring with identity, J(R) is the Ja- cobson radical of R and Z(R) denotes the set of all zero-divisors of R. We denote Fq for the finite field of order q and for any finite subset Y of R, we denote |Y | for the cardinality of Y . The zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the nonzero zero-divisors of R with two distinct vertices a and b joined by an edge if and only if ab = 0. One might ask which graphs on n vertices can be realized as the zero-divisor graph of a com- mutative ring? This question has been partially answered. [1] determines, up to isomorphism, all such rings for which Γ(R) is a graphs on n = 1,2,3, or 4 vertices. This list was extended to n = 5 vertices in [10], and to n = 6, 7, . . . , 14 vertices in [11]. The aim of the paper is to develop this list to a wider class of numbers n. In fact, this observation motivates us the ∗Corresponding author. Email addresses: mbehbood� .iut.a .ir (M. Behboodi), r_beyranvand�math.iut.a .ir (R. Beyranvand) http://www.ejpam.com 303 c© 2010 EJPAM All rights reserved. M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 304 following fundamental question: Question. If R is a finite commutative ring, can we find the relationship between the prime factorizations |Z(R)|= p1 k1 · · · pn kn and the summands Ri, where R= R1×R2× . . .×Rm (m ≥ 1) and R , i s are local rings? Then we will give an answer to this question. We show that the answer is “yes” and a preliminary answer is given in Theorem 1 of Section 2; which shows that if R is a finite commutative ring, then either R is a reduced ring or there are positive integers s, m, t1, . . . , ts, prime numbers p1, p2, . . . , ps and a non-negative integer t such that |Z(R)|= p1 t1 p2 t2 . . . ps ts m and R∼= R1× . . .×Rs× Fq1 × . . .× Fqt with |Z(Ri)| = p ti i . Therefore, in classifying commutative rings with p1 k1 . . . pn kn zero-divisors it suffices to deal with local rings with pi ti zero-divisors where that 1 ≤ i ≤ n and 0 ≤ t i ≤ ki , and henceforth we focus on rings R with |Z(R)| = pk where p is a prime number and k ≥ 1. It is shown that a finite commutative ring R is local if and only if |Z(R)| = pk and |R| = pn for some prime number p and n > k ≥ 0 (Theorem 3). In Section 3, first we characterize commutative rings R with |Z(R)|= pk where 1≤ k ≤ 7. Then the structure and classification up to isomorphism all commutative rings R with |Z(R)|= p1 k1 . . . pn kn , where n ∈ N, p , i s are distinct prime numbers and 1 ≤ ki ≤ 3, are determined. Finally, we determine the structure of nonlocal rings R with |Z(R)|= pk where k = 4 or 5. 2. On Rings with pk Zero-Divisors Recall that an Artinian commutative ring R is called completely primary if R/J(R) is a field. One can easily see that an Artinian commutative ring R is completely primary if and only if Z(R) is an ideal of R, if and only if R is a local ring. Moreover, we have the following lemma which is essentially Theorem 2 of [9]. Lemma 1. [9, Theorem 2] Let R be a finite completely primary ring. Then (i) Z(R) = J(R); (ii) |Z(R)|= p(n−1)r and |R|= pnr for some prime number p, and some positive integers n, r; (iii) Z(R)n = (0); (iv) char(R) = pk for some integer k with 1≤ k ≤ n; (v) R/J(R)∼= Fq, where q = pr . Let Ri (1 ≤ i ≤ s) be a finite commutative ring with mi elements and ni zero-divisors. Let R= R1× . . .×Rs. Then by [6, Theorem 2], |Z(R)|= m1m2 . . . ms−(m1−n1)(m2−n2) . . . (ms− ns). Thus by using this fact, Lemma 1 and the fact that every finite commutative ring is uniquely expressible as a direct sum of completely primary (local) rings (see for example [8, p.95]), we have the following evident result. M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 305 Lemma 2. Let R be a finite commutative ring. Then R ∼= R1 × . . . × Rs where s ∈ N and Ri ,s are local rings with |Ri| = p ki i , |Z(Ri)| = p ti i for some prime numbers p1, p2, . . . , ps and ki ≥ 1, t i ≥ 0. Consequently, |Z(R)|= s ∏ i=1 p ti i ( s ∏ i=1 p ki−ti i − s ∏ i=1 (p ki−ti i − 1)). Now we are in position to prove the following two theorems which are crucial in our investigation. Theorem 1. Let R be a finite commutative ring. Then (i) if R is reduced, then there are finite fields Fq1 , . . . , Fqt (t ≥ 1) such that R ∼= Fq1 × . . . × Fqt with |Z(R)|= q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1). (ii) if R is not reduced, then there are a positive integer s, a non-negative integer t, prime numbers p1, p2, . . . , ps and positive integers k1, . . . , ks such that |Z(R)|= s ∏ i=1 p ti i [q1 . . . qt s ∏ i=1 p ki−ti i − (q1− 1) . . . (qt − 1) s ∏ i=1 (p ki−ti i − 1)] (1) and R∼= R1× . . .×Rs× Fq1 × . . .× Fqt where each Fqi is a finite field and each Ri is a finite local ring such that |Z(Ri)| = p ti i for some 1≤ t i ≤ ki . Consequently, for each i = 1, . . . , s, |Z(Ri)| is a divisor of |Z(R)|. Proof. The proof is clear by Lemma 1 and Lemma 2. Theorem 2. Let R be a commutative ring such that |Z(R)|= pk for some prime number p and a positive number k. Then either (i) R is local, (ii) R is reduced, or (iii) k ≥ 3 and R∼= R1× . . .×Rs× Fq1 × . . .× Fqt where s and t are positive integers, each Fqi is a field, and where each Ri is a commutative finite local ring with |Z(Ri)| = pti , |Ri| = pki for some positive integers ki and t i with 1≤ ∑s i=1 t i ≤ ∑s i=1 ki − s ≤ k− s− 1 such that pk−Σs i=1 ti = q1 . . . qt p Σs i=1(ki−ti) − (q1− 1) . . . (qt − 1)Πs i=1(p ki−ti − 1). (2) Consequently, in the latter case, t i ≤ k − 2 for each i = 1, . . . , s and q j ≡ 1 (p) for some j. Moreover, if t i = k− 2 for some i, then s = t = 1, i.e., R∼= R1× Fq where |Z(R1)|= pk−2 and so p2 = p+ q− 1. M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 306 Proof. Suppose R is not local. Then R ∼= R1 × . . . × Rn, where n ≥ 2 and each Ri is a local ring. If for each i (1 ≤ i ≤ n) Ri is not field, then |Z(Ri)| = pti and |Ri| = pki for some 1≤ t i < ki ≤ k. By the relation (1) of Theorem 1, we have pk = p ∑s i=1 ti[p ∑s i=1(ki−ti) − s ∏ i=1 (pki−ti − 1)] and hence pk− ∑s i=1 ti = p ∑s i=1(ki−ti)− s ∏ i=1 (pki−ti − 1). This implies that 0≡ 1(p) or 0≡ −1(p), a contradiction. Thus R j is field for some 1≤ j ≤ n. If each Ri is field, then R is a reduced ring. Suppose R is non-reduced. Without loss of generality we can assume that R∼= R1 × . . .× Rs × Fq1 × . . .× Fqt where s, t ≥ 1 and each Ri is a commutative finite local ring with |Z(Ri)| = pti and |Ri| = pki for some 1≤ t i < ki ≤ k. Since t ≥ 1, it is easy to check that pk = |Z(R)|> s ∏ i=1 |Ri| = p ∑s i=1 ki ≥ p ∑s i=1(ti+1) = p( ∑s i=1 ti)+s. Consequently we have 1≤ ∑s i=1 t i ≤ ∑s i=1 ki− s ≤ k− s−1 and hence we obtain relation (2). Now since k− ∑s i=1 t i and ∑s i=1(ki− t i) are positive, the relation (2) shows that qi ≡ 1(p) for some i. Also, since s ≥ 1, t i ≤ k− 2 for each i = 1, . . . , s. If t j = k− 2 for some j ∈ {1, . . . , s}, then s = 1. Thus R ∼= R1 × Fq1 × . . .× Fqt , where R1 is a local ring with |Z(R1)| = pk−2. Since |Z(R)|= pk and t ≥ 1, by Theorem 1, |R1| = pk−1. Also by the relation (2) we have p2 = pq1q2 . . . qt − (p− 1)(q1− 1)(q2− 1) . . . (qt − 1). Since qi ≡ 1 (p) for some i, we can assume that q1 ≡ 1 (p) and so q1 > p. Now if t ≥ 2, then |Z(R)| ≥ |R1|q1 > pk−1p = pk, a contradiction. Thus t = 1 and p2 = pq1− (p−1)(q1−1), i.e., p2 = p+ q1 − 1. Obviously for every finite local ring R we have |R| = pn for some prime number p and n ≥ 0. In general, the converse is not true (the nonlocal ring F2 × F2 has 4 elements). Here we show that a finite ring R is local if and only if |Z(R)| = pm and |R| = pn for some prime number p and n> m≥ 0. Theorem 3. Let R be a commutative ring. Then R is a finite local ring if and only if |Z(R)|= pk and |R|= pn for some prime number p and n> k ≥ 0. Proof. For one direction, the proof is clear by Lemma 1. For the other direction, suppose that |Z(R)| = pk and |R| = pn for some prime number p and n > k ≥ 0. If R is not a local ring, then by Theorem 2, either R∼= Fq1 × . . .× Fqt (when R is reduced) or R∼= R1× . . .×Rs× Fq1 × . . . × Fqt where s and t are positive integers and each Ri is a commutative finite local M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 307 ring that is not a field, and where each Fqi is a field. Since |R| = pn, each qi is a divisor of pn and since qi is a prime power, qi ≡ 0 (p) for each i (1 ≤ i ≤ t). If R is reduced, then we have pk = q1 . . . qt − (q1− 1) . . . (qt − 1) and if R is not reduced, then we have pk−Σs i=1 ti = q1 . . . qt p Σs i=1(ki−ti) − (q1− 1) . . . (qt − 1)Πs i=1(p ki−ti − 1). Thus in any case 0≡ 1(p) or 0≡ −1(p), which is impossible. Thus R is a local ring. 3. On Commutative Rings with p1 k1 . . . pn kn(1 ≤ ki ≤ 7) Zero-Divisors By Lemma 1, for each finite local ring R we have |Z(R)| = pk for some prime number p and k ≥ 0, but the converse is not true in general. For example, the nonlocal ring Z8× F7 has 32 zero-divisors. In this section, we will characterize rings with pk zero-divisors where k is a positive integer 1≤ k ≤ 7. Theorem 4. Let R be a commutative ring with |Z(R)| = pk where p is a prime number and 1≤ k ≤ 6. Then either (i) R is a local ring; (ii) R is a reduced ring and so R ∼= Fq1 × . . .× Fqt , where each Fqi (1 ≤ i ≤ t) is a finite field and pk = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1); (iii) R ∼= R1 × Fq1 × . . .× Fqt , where each Fqi (1 ≤ i ≤ t) is a finite field and R1 is a local ring with |Z(R1)|= pm, |R1| = pn such that 0< m< n≤ k− 1 and pk = pnq1q2 . . . qt − (p n− pm)(q1− 1)(q2− 1) . . . (qt − 1); or (iv) R∼= R1 × R2× F5 where each Ri is isomorphic to Z4 or Z2[x]/(x 2). Proof. Suppose |Z(R)| = pk and R is not a local ring. If R is reduced, then we are done. Now let R is not a reduced ring. Then by Theorem 2, we can assume that R∼= R1 × . . .× Rs × Fq1 × . . .× Fqt , where s, t ≥ 1 and each Ri is a local ring with |Z(Ri)| = pti , |Ri| = pki for some t i , ki ≥ 1 such that 1≤ s ∑ i=1 t i ≤ s ∑ i=1 ki − s ≤ k− s− 1≤ 6− 1− 1= 4. It follows that s ≤ 4. If s = 3 or 4, then since t ≥ 1, pk = |Z(R)| > |R1||R2||R3| ≥ p6, this is a contradiction. Hence s ≤ 2. If s = 1, then by Theorem 2, we are done. Thus we can assume that s = 2, i.e., R∼= R1×R2× Fq1 × . . .× Fqt , where R1 and R2 are local rings with |Z(Ri)| = pti . Clearly |Ri| ≥ pti+1 for i = 1, 2 and since t ≥ 1, |Z(R)| > |R1||R2|. If t i ≥ 3 for some i or t1 = t2 = 2, then |Z(R)| > |R1||R2| = pt1+t2+2 ≥ p6, a contradiction. Thus without loss of generality we can assume that either t1 = 2, t2 = 1 or t1 = t2 = 1. M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 308 • Case 1: t1 = 2, t2 = 1 i.e., |Z(R1)| = p2 and |Z(R2)|= p. Then by Lemma 1, we conclude that |R1| = p3 or |R1| = p4 and |R2| = p2. If |R1| = p4, then |Z(R)|> p6, a contradiction. Thus |R1| = p3 and |R2| = p2 and so |Z(R)| > |R1||R2| ≥ p5 i.e., k = 6. We claim that t = 1, for if not, since qi > p for some i (see Theorem 2), |Z(R)| ≥ |R1||R2||Fqi | > p6, a contradiction. Thus t = 1 and hence by using the relation (2) we have p3 = p2q1 − (p− 1)2(q1− 1). This implies that q1(2p−1) = p3− p2+2p−1 and so (2p−1) is a divisor of p2(p−1). But since (2p− 1, p2) = 1, 2p− 1 is a divisor of p− 1, a contradiction. • Case 2: t1 = t2 = 1 i.e., |Z(R1)| = |Z(R2)| = p. Then |Z(R)| > |R1||R2| ≥ p4, i.e., k ≥ 5. If t ≥ 3, then by the relation (2), p2 is a divisor of (qi − 1)(q j − 1) for some 1≤ i, j ≤ t. It follows that qiq j > p2 and hence |Z(R)| > |R1||R2|qiq j > p4p2 = p6, a contradiction. Therefore t ≤ 2. We claim that t = 1. If t = 2, then by the relation (2) we have pk−2 = p2q1q2− (p− 1)2(q1− 1)(q2− 1). (3) Since k ≥ 5, p2 is a divisor of (q1 − 1)(q2− 1). If p2 is a divisor of qi − 1, then qi > p2 and so |Z(R)|> p6, a contradiction. Thus p is a divisor of both q1−1 and q2−1. Hence q1 − 1 = k1p and q2 − 1 = k2p for some positive integers k1 and k2. Then one obtains from (3), pk−4 = (k1p+ 1)(k2p+ 1)− (p− 1)2k1k2, and hence pk−4− (k1 + k2 + 2k1k2)p+ k1k2 − 1= 0. If k = 5, then p = k1k2−1 k1+k2+2k1k2−1 , a contradiction. Thus we can assume that k = 6 and hence p2 − (k1 + k2 + 2k1k2)p+ k1k2 − 1= 0. (4) Thus the equation (4) shows that the integer p is a solution of X 2− (k1 + k2 + 2k1k2)X + k1k2 − 1= 0. (5) Now let µ be another solution of (5). Clearly µ 6= 1, pµ = k1k2 − 1 > 0 and p + µ = k1 + k2 + 2k1k2. It follows that µ is an integer≥ 2 and hence pµ ≥ p + µ, i.e., k1k2 − 1 > k1 + k2 + 2k1k2, a contradiction (since k1, k2 ≥ 1). Thus t = 1 and since |Z(R1)|= |Z(R2)|= p, we have p4 = p2q1− (p− 1)2(q1 − 1) and so q1(2p− 1) = p4 − p2 + 2p− 1. Thus 2p− 1 is a divisor of p2 − 1, i.e., p2 − 1 = (2p−1)a for some positive integer a. Then the equation p2−2ap+a−1= 0 implies that p is a divisor of a−1, i.e., a−1= pλ for some non-negative integer λ. It follows that p and λ are solutions of x2− 2ax + a− 1= 0 and so p+λ = 2a. If λ= 1, then p = a− 1 and p+1= 2a and hence a = 0, a contradiction. Also, if λ > 1, then pλ ≥ p+λ and so a ≤ −1, a contradiction. M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 309 Finally, if λ = 0 , then p = 2, which yields q1 = 5, i.e., R ∼= R1 × R2 × F5 where R1 and R2 are local rings of order 4 with 2 zero-divisors. Now by [2, page 687], each Ri is isomorphic to Z4 or Z2[x]/(x 2). Corollary 1. Let R be a commutative ring with |Z(R)| = p, where p is a prime number. Then R is isomorphic to one of the rings Zp2 , Zp[x]/(x 2) or Fq1 × . . .× Fqt where p = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1). Proof. If R is a local ring, then by Lemma 1, |R| = p2 and hence by [2, page 687], R is isomorphic to Zp2 or Zp[x]/(x 2). If R is not local, then by Theorem 4, R ∼= Fq1 × . . . × Fqt where t ≥ 2 and p = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1). In [5], it was shown that any commutative ring R with m zero-divisors has m2 or fewer elements. It was proved in [7] that if |Z(R)| = m and |R| = m2, then m = pr for some integer r ≥ 1 and some prime p. These rings were categorized in [4] by the use of two constructions. When the ring R is commutative with 1, then there are only two such rings (up to isomorphism) for m = pr : Fpr [x]/(x2) and Zp2[x]/( f (x)), where f (x) is an irreducible polynomial of degree r over Fp. The rings from the second construction in [4] are shown by Raghavendran [9] to all be isomorphic to the ring Zp2[x]/( f (x)) given above, which is called the Galois Ring of order p2r and characteristic p2, denoted GR(p2r , p2). Let p be a prime number. We write Σm for a set of coset representatives of (F∗p) m in F∗p , and Σ0 m = Σm ∪ {0}. Since F∗p is cyclic, |Σm| = (m, p− 1). Corollary 2. Let R be a commutative ring with |Z(R)| = p2, where p is a prime number. Then R is isomorphic to one of the rings Zp3 , Fp[x , y]/(x , y)2, Fp[x]/(x 3), Zp2[x]/(px , x2 − ǫp) where ǫ ∈ Σ0 2, Fp2[x]/(x2), the Galois ring GR(p4, p2) or Fq1 × . . .× Fqt where t ≥ 2 and p2 = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1). Proof. Suppose that R is a local ring with |Z(R)| = p2. Then by Lemma 1, |R| = p3 or p4. If |R| = p3, then by [2, p.687], R is isomorphic to one of the rings Zp3 , Fp[x , y]/(x , y)2, Fp[x]/(x 3), Zp2[x]/(px , x2 − ǫp) where ǫ ∈ Σ0 2. If |R| = p4, then by [9, Theorem 12], R is isomorphic to the Galois ring GR(p4, p2) or Fp2[x]/(x2). Now suppose that R is not a local ring. Then by Theorem 2, R∼= Fq1 × . . .× Fqt with p2 = q1q2 . . . qt− (q1−1)(q2−1) . . . (qt−1). If R is a finite ring then its additive group is a finite abelian group and is thus a direct product of cyclic groups. Suppose these have generators a1,...,an of orders m1,...,mn. Then the ring structure is determined by the n2 products aia j = n ∑ k=1 wi jkak with wi jk ∈ Zmk and thus by the n3 structure constants wi jk for 1≤ i, j, k ≤ n. Thus we introduce a convenient notation, for giving the structure of a finite ring. A presentation for a finite ring R consists of a set of generators a1, a2, . . . , an of the additive group of R together with relations. The relations are of two types: M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 310 (i) miai = 0 for i = 1, ..., n indicating the additive order of ai, and (ii) aia j = ∑n k=1 wi jkak with wi jk ∈ Zmk for 1≤ i, j ≤ n. If the ring R has the presentation above we write R= D a1, . . . , an; miai = 0, aia j = n ∑ k=1 wi jkak, for i, j = 1, . . . , n E . Corollary 3. Let R be a commutative ring with |Z(R)|= p3, where p is a prime number. Then R is isomorphic to one of the rings Zp2× Fq , Zp[x]/(x 2)× Fq where p2 = p+q−1, Fq1 × . . .× Fqt where p3 = q1q2 . . . qt − (q1 − 1)(q2 − 1) . . . (qt − 1), the Galois ring GR(p6, p2), Fp3[x]/(x2), Fp[x]/(x 4), Zp2[x]/(px , x3), Zp2[x]/(x2), Zp2[x]/(px , x3−ap) where a ∈ Σ3, Zp2[x]/(x2− bp) where b ∈ Σ2 and p 6= 2, Z4[x]/(x 2 − 2), Z4[x]/(x 2 − 2x − 2), Zp2[x , y]/(p, x , y)2, Fp[x , y, z]/(x , y, z)2, Zp3[x]/(px , x2− cp2) where c ∈ Σ0 2, Z4[x]/(x 2− 2x), Zp4 , R1 := 〈1, x1, x2, y; p1 = 0, x1 2 = y, x2 2 = 0, x i x j = x i y = y x i = y2 = 0, i 6= j〉, R2 := 〈1, x1, x2, y; p1 = 0, x1 2 = y, x2 2 = y, x i x j = x i y = y x i = y2 = 0, i 6= j〉, R3 := 〈1, x1, x2, y; p1 = 0, x1 2 = y, x2 2 = ξy, x i x j = x i y = y x i = y2 = 0, i 6= j〉, R4 := 〈1, x1, x2; p21= px1 = px2 = 0, x1 2 = p, x2 2 = 0, x1 x2 = x2 x1 = 0〉, R5 := 〈1, x1, x2; p21= px1 = px2 = 0, x1 2 = ξp, x2 2 = 0, x1 x2 = x2 x1 = 0〉, R6 := 〈1, x1, x2; p21= px1 = px2 = 0, x1 2 = p, x2 2 = p, x1 x2 = x2 x1 = 0〉, R7 := 〈1, x1, x2; p21= px1 = px2 = 0, x1 2 = p, x2 2 = ξp, x1 x2 = 0〉, where ξ is a non-square in Fp and if p = 2 then instead of R3, R5 and R7, R is isomorphic to R′3 or R′5 where R′3 := 〈1, x1, x2; 4.1= 2x1 = 2x2 = 0, x1 2 = 0, x2 2 = 0, x1 x2 = x2 x1 = 2〉 or R′5 := 〈1, x1, x2, y; 2.1 = 0, x1 x2 = x2 x1 = y, x1 2 = x2 2 = x i y = y x i = y2 = 0〉. Proof. If R is a local ring with |Z(R)| = p3, then by Lemma 1, either |R| = p4 or |R| = p6. If |R| = p6, then by [9, Theorem 12], R is isomorphic to Fp3[x]/(x2) or the Galois ring GR(p6, p2). If |R| = p4, then since |Z(R)| = p3, by using [2, p.687-690], one can easily see that R is isomorphic to one of the local rings in above list. Now suppose that R is not local. Then by Theorem 2, R is isomorphic to one of the rings Zp2 × Fq , Zp[x]/(x 2)× Fq where p2 = p + q − 1 or Fq1 × . . . × Fqt where p3 = q1q2 . . . qt − (q1 − 1)(q2 − 1) . . . (qt − 1). This completes the proof. Now we are in position to determine the structure of commutative rings R with |Z(R)| = p k1 1 p k2 2 . . . p kn n , where n≥ 1, 1≤ ki ≤ 3 and pi ,s are distinct prime numbers. Theorem 5. Let R be a commutative ring with |Z(R)| = p k1 1 p k2 2 . . . p kn n , where n ≥ 1, 1 ≤ ki ≤ 3 and pi ,s are distinct prime numbers. Then there exist 0≤ s ≤ Σn i=1ki and t ≥ 0 such that R∼= R1 × . . .× Rs × Fq1 × . . .× Fqt M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 311 where Fqi ,s are finite fields and each Ri is local ring with |Z(Ri)| = p t j j for some p j (1 ≤ j ≤ n) and 1 ≤ t j ≤ k j. Consequently, each Ri is isomorphic to one of the local rings described in Corollaries 1, 2 or 3. Proof. We put R∼= R1× . . .× Rs × Fq1 × . . .× Fqt , where Fq1 , . . . , Fqt are finite fields and each Ri is a commutative finite local ring that is not a field. If s = 0, then there is nothing to prove. Thus we can assume that s ≥ 1 and so by Theorem 1, for each i, |Z(Ri)| = pk for some prime number p and k ≥ 1 such that pk is a divisor of |Z(R)| and also 0≤ s ≤ Σn i=1ki. Thus |Z(Ri)|= p t j j where 1≤ t j ≤ k j, 1≤ j ≤ n and 1 ≤ i ≤ s. Thus for each 1 ≤ i ≤ s, t i = 1,2 or 3 and so each Ri is isomorphic to one of the local rings described in Corollaries 1, 2 or 3 Next, we determine the structure of commutative nonlocal rings R with |Z(R)|= p4 or p5 where p is a prime number. Proposition 1. Let R be a commutative nonlocal ring with |Z(R)| = p4 where p is a prime number. Then R∼= Fq1 ×. . .×Fqt with p4 = q1q2 . . . qt−(q1−1)(q2−1) . . . (qt−1), R∼= R1×Fq1 × . . .×Fqt where R1 ∼= Zp2 or Zp[x]/(x 2) with p3 = pq1q2 . . . qt−(p−1)(q1−1)(q2−1) . . . (qt−1) or R ∼= R1 × Fq where and p2 = p+ q− 1 and R1 is isomorphic to one of the local rings of order p3 described in Corollary 2. Proof. By Theorem 4, either R is reduced or R ∼= R1 × Fq1 × . . .× Fqt , where R1 is a local ring with |Z(R1)| = p or p2 and t ≥ 1. Now we proceed by cases. • Case 1: R is reduced. Then R∼= Fq1 × . . .× Fqt with p4 = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1). • Case 2: R is not reduced and |Z(R1)| = p. Then by Corollary 1, R1 is isomorphic to Zp2 or Zp[x]/(x 2) and p3 = pq1q2 . . . qt − (p− 1)(q1− 1)(q2− 1) . . . (qt − 1). • Case 3: R is not reduced and |Z(R1)| = p2. Then by Lemma 1, either |R1| = p3 or |R1| = p4. Since t ≥ 1, p4 = |Z(R)|> |R1| and hence |R1| = p3. Thus by Corollary 2, R1 is isomorphic to one of the rings Fp[x , y]/(x , y)2, Fp[x]/(x 3), Zp3 or Zp2[x]/(px , x2−ǫp) where ǫ ∈ Σ0 2 with p2 = p+ q− 1. Proposition 2. Let R be a commutative nonlocal ring with |Z(R)| = p5 where p is a prime number. Then (i) R∼= Fq1 × . . .× Fqt with p5 = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1), (ii) R ∼= R1 × Fq1 × . . . × Fqt where R1 ∼= Zp2 or Zp[x]/(x 2) with p4 = pq1q2 . . . qt − (p − 1)(q1− 1)(q2− 1) . . . (qt − 1), M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 312 (iii) R ∼= R1 × Fq1 × . . .× Fqt where R1 is isomorphic to one of the rings Zp3 , Fp[x , y]/(x , y)2, Fp[x]/(x 3), or Zp2[x]/(px , x2− ǫp) where ǫ ∈ Σ0 2 and p3 = pq1q2 . . . qt − (p− 1)(q1− 1)(q2− 1) . . . (qt − 1), (iv) R∼= R1 × Fq where R1 ∼= Fp2[x]/(x2) or GR(p4, p2) with p3 = p2 + q− 1, or (v) R ∼= R1 × Fq and p2 = p+ q− 1 where R1 is isomorphic to one of the local rings of order p4 described in Corollary 3. Proof. By Theorem 4, either R is reduced or R ∼= R1 × Fq1 × . . .× Fqt , where R1 is a local ring with |Z(R1)| = p, p2 or p3 and t ≥ 1. Now we proceed by cases. • Case 1: R is reduced. Then R∼= Fq1 × . . .× Fqt with p5 = q1q2 . . . qt − (q1 − 1)(q2− 1) . . . (qt − 1) • Case 2: R is not reduced and |Z(R1)| = p. Then by Corollary 1, R1 is isomorphic to Zp2 or Zp[x]/(x 2) and p4 = pq1q2 . . . qt − (p− 1)(q1− 1)(q2− 1) . . . (qt − 1). • Case 3: R is not reduced and |Z(R1)| = p2. Then by Lemma 1, |R1| = p3 or p4. If |R1| = p3, then by Corollary 2, R1 is isomorphic to one of the ring Zp3 , Fp[x , y]/(x , y)2, Fp[x]/(x 3), or Zp2[x]/(px , x2− ǫp) where ǫ ∈ Σ0 2 and p3 = pq1q2 . . . qt − (p− 1)(q1− 1)(q2− 1) . . . (qt − 1). If |R1| = p4, then t = 1, for if not, then by Theorem 2, qi > p for some i and hence |Z(R)| > |R1|qi > p5, a contradiction. Thus t = 1 and so R ∼= R1 × Fq with p3 = p2 + q− 1. Moreover, since |Z(R1)| = p2 and |R1| = p4, by [9, Theorem 12], R1 is isomorphic to Fp2[x]/(x2) or GR(p4, p2). • Case 4: R is not reduced and |Z(R1)| = p3. Then by Lemma 1, |R1| = p4 or p6. If |R1| = p6, then |Z(R)| ≥ p6, a contradiction (we note that t ≥ 1). Thus |R1| = p4 and so by Theorem 2, t = 1, i.e., R ∼= R1 × Fq with p2 = p + q − 1. Now since |R1| = p4 and |Z(R1)| = p3, R1 is isomorphic to one of the local rings of order p4 described in Corollary 3. The next theorem characterizes commutative rings with p7 zero-divisors. Theorem 6. Let R be a commutative ring with |Z(R)| = p7 where p is a prime number. Then either (i) R is a local ring with |R|= p8 or p14; (ii) R is a reduced ring and so R ∼= Fq1 × . . .× Fqt , where each Fqi (1 ≤ i ≤ t) is a finite field and p7 = q1q2 . . . qt − (q1− 1)(q2− 1) . . . (qt − 1); (iii) R ∼= R1 × Fq1 × . . .× Fqt , where each Fqi (1 ≤ i ≤ t) is a finite field and R1 is a local ring with |Z(R1)| = pm, |R1| = pn such that 0< m < n≤ 6 and p7 = pnq1q2 . . . qt − (p n− pm)(q1− 1)(q2− 1) . . . (qt − 1); M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 313 (iv) R ∼= R1 × R2 × Fq1 × . . . × Fqt , where each Fqi (1 ≤ i ≤ t) is a finite field, each Ri is isomorphic to Zp2 or Zp[x]/(x 2) and p5 = p2q1q2 . . . qt − (p− 1)2(q1− 1)(q2− 1) . . . (qt − 1); or (v) R ∼= R1 × R2 × F5 where R1 is isomorphic to one of the rings Z4 or Z2[x]/(x 2) and R2 is isomorphic to one of the rings Z8, Z2[x , y]/(x , y)2, Z2[x]/(x 3), or Z4[x]/(2x , x2− 2ǫ) where ǫ ∈ Σ0 2. Proof. Suppose |Z(R)| = p7 and R is not a local ring. If R is reduced, then we are done. Now suppose R is not a reduced ring. Then by Theorem 2, we can assume that R∼= R1× . . .× Rs × Fq1 × . . .× Fqt , where s, t ≥ 1 and each Ri is a local ring with |Z(Ri)|= pti , |Ri| = pki for some t i , ki ≥ 1 such that 1≤ s ∑ i=1 t i ≤ s ∑ i=1 ki − s ≤ 7− s− 1≤ 7− 1− 1= 5. It follows that s ≤ 5. We claim that s = 1 or 2, for if not either s ≥ 4 or s = 3. If s ≥ 4, then |Z(R)| > p8 (because |Ri| ≥ p2 for all i), a contradiction. Now let s = 3. If |Z(Ri)| = pti with t i ≥ 2 for some i, then |Z(R)| > p7, a contradiction. Thus |Z(Ri)| = p for i = 1,2,3. Now by the relation (1) of Theorem 1, we have p7 = p6q1q2 . . . qt − (p 2 − p)3(q1− 1)(q2− 1) . . . (qt − 1). Thus p4 = p3q1q2 . . . qt − (p− 1)3(q1 − 1)(q2− 1) . . . (qt − 1) and so p is a divisor of (qi − 1) for some i (so qi > p). Now if t ≥ 2, then |Z(R)| ≥ |R1||R2||R3|qi > p7, this is a contradic- tion. Thus t = 1 and so the relation p4 = p3q1 − (p− 1)3(q1 − 1) implies that p3 is a divisor of (q1−1). Hence q1 > p3 and so |Z(R)| ≥ |Z(R1)||R2||R3|q1 > p7, a contradiction. Thus s ≤ 2. Suppose s = 1, i.e., R ∼= R1 × Fq1 × . . .× Fqt . Then by Theorem 2, either R1 is a local ring with |Z(R1)|= pt1 where t1 = 1,2,3 or 4 such that p7 = |R1|q1× . . .× qt − (|R1| − pk)(q1− 1)× . . . (qt − 1) or R∼= R1× Fq1 where |R1| = p6, |Z(R1)| = p5 and p2 − p− q1 + 1= 0. Now suppose s = 2. The proof now proceeds by cases. • Case 1: t1 ≥ 3 or t2 ≥ 3. Without loss of generality we can assume that t1 ≥ 3. Then |R1| = pk1 , and |R2| = pk2 where k1 ≥ 4 and k2 ≥ 2. If k1 ≥ 5 or k2 ≥ 3, then |Z(R)| > |R1||R2| ≥ p7, a contradiction. Now let k1 = 4 and k2 = 2. By Theorem 2, qi > p for some 1 ≤ i ≤ t. If t ≥ 2, then |Z(R)| > |R1||R2|qi ≥ p7, a contradiction. Thus t = 1 and so p7 = p6q1 − (p 4 − p3)(p2 − p)(q1 − 1). It follows that p2 is a divisor of q1 − 1 and so q1 > p2. Hence |Z(R)|> |Z(R1)||R2||Fq1 |> p7, a contradiction. M. Behboodi, R. Beyranvand / Eur. J. Pure Appl. Math, 3 (2010), 303-316 314 • Case 2: t1 = t2 = 2 i.e., |Z(R1)| = |Z(R2)| = p2. Then by Lemma 1, |R1| = pk1 and |R2| = pk2 where 3 ≤ k1, k2 ≤ 4. If k1 = 4 or k2 = 4, then |Z(R)| > |R1||R2| ≥ p7, a contradiction. Now let k1 = k2 = 3. By Theorem 2, qi > p for some 1 ≤ i ≤ t. If t ≥ 2, then |Z(R)| > |R1||R2|qi ≥ p7, a contradiction. Thus t = 1 and so p7 = p6q1− (p 3− p2)2(q1− 1). It follows that p2 is a divisor of q1− 1 and so q1 > p2. Hence |Z(R)|> |Z(R1)||R2||Fq1 | > p7, a contradiction. • Case 3: t1 = 2 and t2 = 1 or t1 = 1 and t2 = 2. Without loss of generality we can assume that t1 = 2 and t2 = 1. By Lemma 1, either |R1| = p3 or |R1| = p4 and |R2| = p2. By the proof in Case 1, |R1| 6= p4. Thus |R1| = p3 and |R2| = p2 and hence by the relation (2) of Theorem 2 we have p4 = p2q1q2 . . . qt − (p− 1)2(q1− 1)(q2− 1) . . . (qt − 1). (6) It follows that p2 is a divisor of (q1 − 1)(q2 − 1) . . . (qt − 1). Without loss of generality we may assume that p2 is a divisor of (q1 − 1) or p is a divisor of (q1− 1) and (q2 − 1). If t > 2, then |Z(R)| > |R1||R2||Fq1 ||Fq2 | ≥ p7, a contradiction. Thus t ≤ 2 and so the proof now proceeds by subcases. – Subcase 1: t = 1 i.e., R∼= R1× R2 × Fq1 . Then by (6), we have p4 = p2q1 − (p− 1)2(q1− 1). (7) This implies that p2 is a divisor of (q1 − 1). Thus q1 − 1 = p2k for some positive integer k and so by using (7) we obtain p2−2pk+k−1= 0. This equation implies that p is a divisor of k − 1, i.e., k − 1 = pλ for some non-negative integer λ. It follows that p and λ are solutions of x2 − 2kx + k− 1 = 0 and so p+ λ = 2k. If λ = 1, then p = k − 1 and p + 1 = 2k and hence k = 0, a contradiction. Also, if λ > 1, then pλ ≥ p + λ and so k ≤ −1, a contradiction. Finally, if λ = 0, then p = 2 which yields q1 = 5, i.e., R ∼= R1 × R2 × F5 where R1 and R2 are local rings with |R1| = 8 and |R2| = 4. Since |Z(R1)| = 4, by [2, p.687], R1 is isomorphic to one of the rings Z8, Z2[x , y]/(x , y)2, Z2[x]/(x 3), or Z4[x]/(2x , x2 − 2ǫ) where ǫ ∈ Σ0 2. Also, R2 is isomorphic to Z4 or Z2[x]/(x 2). – Subcase 2: t = 2 i.e., R∼= R1× R2 × Fq1 × Fq2 . Then by (6), we have p4 = p2q1q2− (p− 1)2(q1− 1)(q2− 1). Thus p2 is a divisor of (q1− 1)(q2− 1). If p2 is a divisor of q1− 1 (or q2− 1), then q1 > p2 (or q2 > p2) and so |Z(R)| > |R1||R2||Fqi | > p7 where i = 1 or 2, this is a contradiction. Thus p is a divisor of both q1 − 1 and q2 − 1. Hence q1 − 1 = k1p and q2− 1= k2p for some positive integers k1 and k2. Then one obtains from (6), p2 = (k1p+ 1)(k2p+ 1)− (p− 1)2k1k2, and hence p2 − (k1 + k2 + 2k1k2)p+ k1k2 − 1= 0. (8) REFERENCES 315 Now the equation (8) shows that the integer p is a solution of X 2− (k1 + k2 + 2k1k2)X + k1k2 − 1= 0. (9) Now let µ be another solution of (9). Clearly µ 6= 1, pµ = k1k2 − 1 > 0 and p+ µ = k1 + k2 + 2k1k2. It follows that µ is an integer≥ 2 and hence pµ ≥ p+µ, i.e., k1k2 − 1> k1 + k2 + 2k1k2, a contradiction (since k1, k2 ≥ 1). • Case 4: t1 = t2 = 1 i.e., |Z(R1)| = |Z(R2)| = p and R ∼= R1 × R2 × Fq1 × . . .× Fqt with p5 = p2q1q2 . . . qt − (p − 1)(q1 − 1)(q2 − 1) . . . (qt − 1). On the other hand by by [2, p.687], R1 is isomorphic to Zp2 or Zp[x]/(x 2). Finally, we conclude the article with the next remark which is a good justification for the classification up to isomorphism commutative rings with p1 k1 . . . pn kn zero-divisors, where 1≤ ki ≤ 5. Remark 1. We remark that by Propositions 1 and 2, for classifying up to isomorphism commu- tative rings with p1 k1 . . . pn kn zero-divisors, where 1 ≤ ki ≤ 5, it suffices to find local rings with |R|= p6, |Z(R)|= p4 and |R|= p6, |Z(R)|= p5. In fact, if R is a local ring with |Z(R)|= p4, then by Lemma 1, |R|= p5, p6 or p8. The local rings of order p5 is determined in [3]. Also, if |R|= p8, then by [9, Theorem 12], R is isomorphic to the Galois ring GR(p8, p2) or Fp4[x]/(x2). On the other hand, if R is a local ring with |Z(R)|= p5, then by Lemma 1, |R|= p6 or p10. If |R|= p10, then by [9, Theorem 12], R is isomorphic to the Galois ring GR(p10, p2) or Fp5[x]/(x2). ACKNOWLEDGEMENTS This work was partially supported by IUT (CEAMA). The research of the first author was in part supported by a grant from IPM (No. 87160026). References [1] D.F. Anderson, A. Farzier, A. Lauve, and P.S. Livingston, The zero-divisor graph of a commutative ring, II, Lecture notes in pure and appl. Math. 202 (2001), 61-72, Marcel Dekker, New York. [2] B. Corbas and G.D. Williams, Rings of order p5 Part I. Nonlocal rings, J. Algebra, 231 (2000), 677-690. [3] B. Corbas and G.D. Williams, Rings of order p5 Part II. Local rings, J. Algebra, 231 (2000), 691-704. [4] B. Corbas, Rings with few zero-divisors, Math. Ann. 181 (1) (1969), 1Ű7. [5] N. 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