Jay Shapiro

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

Abstract For a commutative ring R with set of zero-divisors Z ( R ), the zero-divisor graph of R is Γ ( R )= Z ( R )−{0}, with distinct vertices x and y adjacent if and only if xy =0. In this paper, we show that Γ ( T ( R )) and Γ ( R ) are isomorphic as graphs, where T ( R ) is the total quotient ring of R , and that Γ ( R ) is uniquely complemented if and only if either T ( R ) is von Neumann regular or Γ ( R ) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.

THE ZERO-DIVISOR GRAPH OF VON NEUMANN REGULAR RINGS

Let R be a commutative ring, and let ZðRÞ denote its set of zerodivisors. We associate a simple graph GðRÞ to R with vertices ZðRÞ 1⁄4 ZðRÞ f0g, the set of nonzero zero-divisors of R. Two distinct vertices x and y are adjacent if xy 1⁄4 0. Thus GðRÞ is the empty graph if and only if R is an integral domain. The notion of a zero-divisor graph was first introduced by I. Beck in [4] and further investigated in [1], where the authors were interested in colorings of GðRÞ, though their vertex set included the zero element. In [2, 3, 5] and [7] the authors, using the same vertex set as here, were interested in examining the interplay between the ring-theoretic properties of R and the graph-theoretic properties of GðRÞ. We examine the zero-divisor graph GðRÞ, of a Von Neumann regular ring R. Recall that a ring R is called Von Neumann regular if for each a 2 R there exists b 2 R such that ab 1⁄4 a, or equivalently if every principal ideal of R is generated by an idempotent. We show that if R and S are two such rings with GðRÞ ffi GðSÞ, then they share certain ring-theoretic properties. If the socle is essential in one, then it is essential in the other (Proposition 2.5). In Theorem 2.3 we show that if the socle of one is the direct sum of finite simple ideals, then so is the other and the socles are isomorphic as rings (without identities). In Corollary 2.4 we show that if two rings are the

Mixed matrices and binomial ideals

Given an integral vector u /gE Zn, one may associate with it the binomial /tfu = Xu+ − Xu− in Z[X] = Z[X1, …, Xn] where u+ and u− are the positive and negative supports of u, respectively. We say that u is mixed if u+,u− ≠ 0 and a matrix M is mixed if all its rows are mixed. We investigate relationships between the matrix M whose rows are u1, …, ur and the ideal I = 〈/tfu1, …, /tfu1〉. For example, if M contains no square mixed submatrix of any size, then the vectors u1, …, ur are linearly independent and /tfu1, …, /tfur form a regular sequence in Z[X]. This allows us to decide if a semigroup ring is a complete intersection. When applied to numerical semigroups, the results give an alternate proof of a theorem by Delorme which characterizes numerical semigroups that are complete intersections.

Affine semigroup rings that are complete intersections

This paper presents a result concerning the structure of affine semigroup rings that are complete intersections. It generalizes to arbitrary dimensions earlier results for semigroups of dimension less than four. The proof depends on a decomposition theorem for mixed dominating matrices.

Mixed dominating matrices

Abstract We characterize the class of matrices for which the set of supports of nonnegative vectors in the null space can be determined by the signs of the entries of the matrix. This characterization is in terms of mixed dominating matrices, which are defined by the nonexistence of square submatrices that have nonzeros of opposite sign in each row. The class of mixed dominating matrices is contained in the class of L -matrices from the theory of sign-solvability, and generalizes the class of S -matrices. We give a polynomial-time algorithm to decide if a matrix is mixed dominating. We derive combinatorial conditions on the face lattice of a Gale transform of a matrix in this class.

Homological dimensions in a Morita context with applications to subidealizers and fixed rings

Given a Morita context (R, S, V, W, 0, V/), we investigate the relationship between the various homological dimensions of the rings R and S. We then apply these results to two particular examples: subidealizers and fixed rings. Let R and S be rings, RVSKI SWR bimodules and 0: V Os W -* R and qi: W R V -* S bimodule homomorphisms. Consider the array T R VA If 0 and vy satisfy the associativity conditions required to make T a ring, then we call the collection (R, S, V, W, 0, yV) a Morita context and T the ring of the Morita context (see [McCR] for details). Recently, in [C] and [LS], Morita contexts have been used as a tool to relate properties of a ring R and its fixed subring R G, where G is a group of automorphisms of R. In this paper we also use Morita contexts as a device to obtain information on the homological dimensions of fixed rings and subidealizers. In the first section, we relate the various homological dimensions of the rings R and T (in a Morita context) under certain assumptions on V or V and W (Lemma 1.2 and Theorem 1.8). If, in addition 0 is onto, it is known that T and S are Morita equivalent (see [LS, Lemma 1.2]). Thus we have a relationship between the homological dimensions of R and S. In ?2, we apply these results to two naturally occurring Morita contexts. In the first example we show that if S is a subidealizer of a generative right ideal I of R (i.e., S is a subring of R containing I as a two-sided ideal and RI = R then w. gldim(R) < w. gldim(S) < max{/B + 1, w. gldim(R)}, where ,B = w. gldim(S/I) + wdim(R/I)R (Theorem 2.1), and w. gldim(X) denotes the weak global dimension of the ring X. Robson and Small [RS] have Received by the editors December 7, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A26, 16A60. ( 1990 American Mathematical Society 0002-9939/90

Universal Lying-Over Rings

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Localization in a Morita context with applications to fixed rings

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The Ohm–Rush content function

A (commutative unital) ring R is said to satisfy universal lying-over (ULO) if each injective ring homomorphism R → T satisfies the lying-over property. If R satisfies ULO, then R = tq(R), the total quotient ring of R. If a reduced ring satisfies ULO, it also satisfies Property A. If a ring R = tq(R) satisfies Property A and each nonminimal prime ideal of R is an intersection of maximal ideals, R satisfies ULO. If 0 ≤ n ≤ ∞, there exists a reduced (resp., nonreduced) n-dimensional ring satisfying ULO. The A + B construction is used to show that if 2 ≤ n < ∞, there exists an n-dimensional reduced ring R such that R = tq(R), R satisfies Property A, but R does not satisfy ULO.

A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

In the study of a noetherian ring R, there are two related questions that have played a major role in recent years; one is finding prime ideals of R which satisfy the second layer condition, the other is finding the (second layer) link closed sets of prime ideals of R. In particular these properties are necessary to determine the classical sets of prime ideals of R, which allows one to obtain a well behaved Ore localization (see, for example, [24, 8, 91). One aspect of this problem is the transfer of these properties from the prime spectrum of one ring R to the prime spectrum of another, related ring, A. Jategaonkar [S] has shown that, if R is a commutative noetherian ring, then the group ring RG satisfies the second layer condition whenever G is polycyclic-by-finite. This result was then generalized by Bell [2] to strongly group graded rings. More recently Letzter [lo] has studied the relationship between the prime spectrums of the noetherian rings R and S, when S is module finite over R, showing how the second layer condition and links between prime ideals transfer from one ring to the other. Let T be a right noetherian ring and let e be an idempotent element of T, then it is well known that eTe is a right noetherian ring. In Sections 1 and 2 we examine the transfer of the second layer condition between the prime ideals of T and eTe, as well as the transfer of classical sets of prime ideals between these rings. This enables us to study how these properties of Spec(eTe) translate to Spec(( 1 e) T( 1 e)). In particular, under certain hypotheses, we prove that whenever eTe satisfies the second layer condition, then so does (1 e) T( 1 e) (Corollary 2.6). In addition we are able to show that under the same hypotheses, if every prime ideal of eTe belongs to a finite classical set, then so does every prime ideal of (1 -e) T(l -e). Another way to describe the relationship between T and eTe is via a Morita context (see [ 11). Let R and S be rings, let s W, and R V, be bimodules, and let 6: VOs W-+ R,