Mediterr. J. Math. 6 (2010) 139–150DOI 10.1007/s00009-013-0276-y© 2013 Springer Basel

Mediterranean Journalof Mathematics

On the Strong (A)-Rings of Mahdou andHassani

David E. Dobbs∗ and Jay Shapiro

Abstract. A (commutative unital) ring R with only finitely many mini-mal prime ideals (for instance, a Noetherian ring) is reduced and astrong (A)-ring if and only if R is an integral domain. Thus, the smallestreduced ring which has Property A but is not a strong (A)-ring is Z2×Z2.A Noetherian ring R is a strong (A)-ring if and only if AssR(R) has aunique maximal element.

Mathematics Subject Classification (2010). Primary 13A15; Secondary13G05, 13C13, 13E05, 13F10.Keywords. Commutative ring, Property A, zero–divisor, annihilator,prime ideal, associated prime, Noetherian, finite ring, integral domain.

1. Introduction

In this note, all rings are commutative, unital and nonzero. If R is a ring, itsset of zero-divisors is denoted by Z(R); and if E is an R-module, then Ann(E)denotes the annihilator of E. As in [1], a ring R is said to have Property Aif, whenever I is a finitely generated ideal of R such that I ⊆ Z(R), we havethat Ann(I) �= 0. It is well known that any Noetherian ring has Property A(cf. [2, Theorem 82]). Recently, Mahou and Hassani [3] have defined a strong(A)-ring to be a ring R such that if an ideal I of R can be generated by a finitesubset of Z(R), then Ann(I) �= 0. It is clear that any strong (A)-ring musthave Property A. Much of [3] is devoted to proving that the converse is false.Indeed, [3, Theorem 2.1] characterizes when an idealization satisfies PropertyA or is a strong (A)-ring and then uses this fact to give a family of infiniterings that satisfy Property A but fail to be strong (A)-rings [3, Example, page397] (that work is built on a certain infinite ring D in [3, Example, page 395]which satisfies Property A but is not a strong (A)-ring). The work that wehave just summarized from [3] covers 2 1/2 pages. The point of the present

∗Corresponding author.

D. E. Dobbs and J. Shapiro

note is that one needs not work so hard in order to construct a natural familyof rings that satisfy Property A but are not strong (A)-rings.

Proposition 2.1 gives the underlying theoretical result, together withits impact in case the ring is Noetherian (or even finite), and Example 2.2gives some natural families of examples and identifies the resulting smallestpossible reduced ring that is an example. Then Proposition 2.3 characterizesthe Noetherian strong (A)-rings, and we close by identifying a non-reducedring R of minimal cardinality such that R satisfies Property A but is not astrong (A)-ring.

2. Results

Recall that a ring is said to be reduced if it has no nonzero nilpotent ele-ments. Clearly, any integral domain is reduced. On the other hand, if D,Eare integral domains, then D × E is a reduced ring which is not an integraldomain.

Proposition 2.1. Let R be a ring with only finitely many minimal prime ideals(for instance, a Noetherian ring or even a finite ring). Then R is reduced anda strong (A)-ring if and only if R is an integral domain.

Proof. It is clear from the definitions that any integral domain is a strong(A)-ring, and it is reduced by the above remark. Conversely, let R be areduced ring which is also a strong (A)-ring. Consider Min(R), the set ofminimal prime ideals of R. Since R is nonzero, Min(R) is nonempty (cf. [2,Theorem 10]). By hypothesis, Min(R) is finite, say with n elements: Min(R) ={P1, . . . , Pn}. Also, since R is reduced, Z(R) = ∪{P | P ∈ Min(R)} [1,Corollary 2.4]. If Min(R) is a singleton set, the unique minimal prime idealof R must be 0 (since ∩{P | P ∈ Min(R)} is the set of nilpotent elementsof R), and it follows that Z(R) = {0}, so that R is an integral domain.Hence, without loss of generality, n ≥ 2. By the Prime Avoidance Lemma [2,Theorem 81], we can pick x ∈ P1 \ ∪n

i=2 Pi (since P1 �⊆ Pi for all i ≥ 2). It iseven easier to pick y ∈ (∩n

i=2 Pi) \ P1 (since Pi �⊆ P1 for all i ≥ 2). Clearly,z := x+ y �∈ ∪n

i=1 Pi = Z(R). But this is a contradiction since (z ∈ Rx+Ryand) the “strong (A)-ring” property gives Rx+Ry ⊆ Z(R). �

Example 2.2. (a) Let R be a finite reduced ring. By Artin-Wedderburn theory,this assumption is equivalent to R being (isomorphic to) the direct productof finitely many (finite) fields: R ∼= ∏n

i=1 Ki, where n is a positive integerand each Ki is a finite field. Suppose that n ≥ 2. Then R is not an integraldomain and so, by Proposition 2.1, R is not a strong (A)-ring. Of course,R has Property A since R is Noetherian. The smallest such R is clearly(isomorphic to) Z2 × Z2.

(b) The argument that was given in (a) also shows that if K1, . . . ,Kn

is any finite list of (possibly infinite) fields and n ≥ 2, then the reducedNoetherian ring

∏ni=1 Ki fails to be a strong (A)-ring (although it does satisfy

Property A).

Vol. 6 (2010) Strong (A)-Rings of Mahdou and Hassani 3

Recall that a prime ideal P of a ring R is said to be an associated primeof an R-module E if P is the annihilator of some e ∈ E. The set of associatedprimes of E is denoted by AssR(E). It is well known (cf. [4, Theorems 6.1 and6.5]) that if R is a Noetherian ring, then AssR(R) is a nonempty finite setand ∪{P | P ∈ AssR(R)} = Z(R). Associated primes figure in the followingcompanion for the “Noetherian” case of Proposition 2.1.

Proposition 2.3. A Noetherian ring R is a strong (A)-ring if and only ifAssR(R) has a unique maximal element.

Proof. If the “only if” assertion fails, choose elements x and y as in the proofof Proposition 2.1, with the maximal elements of AssR(R) playing the roleplayed earlier by the minimal prime ideals of R, and obtain a contradictionby adapting the proof of Proposition 2.1. For the converse, let M denote theunique maximal element of AssR(R). It follows from the above commentsthat Z(R) = M . As M = Ann(Re) for some e ∈ R \ {0}, we see that eannihilates any ideal of R that is generated by a subset of Z(R), and so R isa strong (A)-ring. �

An interesting consequence is that 8 is the smallest cardinality of a non-reduced ring R which has Property A but is not a strong (A)-ring: considerR := Z4 × Z2.

References

[1] J. A. Huckaba, Commutative Rings with Zero Divisors. Dekker, New York, 1988.[2] I. Kaplansky, Commutative Rings. Revised Edition, Univ. Chicago Press,

Chicago, 1974.[3] N. Mahdou and A. R. Hassani, On strong (A)-rings. Mediterr. J. Math. 9 (2012),

393-402.[4] H. Matsumura, Commutative Ring Theory. 2nd Edition, Cambridge Univ. Press,

Cambridge, 1989.

David E. DobbsDepartment of MathematicsUniversity of TennesseeKnoxville, Tennessee 37996-1320U.S.A.e-mail: dobbs@math.utk.edu

Jay ShapiroDepartment of MathematicsGeorge Mason UniversityFairfax, Virginia 22030-4444U.S.A.e-mail: jshapiro@gmu.edu

Received: October 19, 2012.Accepted: January 22, 2013.