This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms of comaximal graphs and the rings in question. In addition, we investigate the relation between the comaximal graph of a ring and its subrings of a certain type.

In this paper we investigate principal prime ideals in commutative rings. Among other things we characterize the principal prime ideals that are both minimal and maximal and characterize the maximal ideals of a polynomial ring that are principal. Our main result is that if (p) is a principal prime ideal of an atomic ring R, then ht(p)≤1.

For a split semisimple Chevalley group scheme G with Lie algebra over an arbitrary base scheme S, we consider the quotient of by the adjoint action of G. We study in detail the structure of over S. Given a maximal torus T with Lie algebra and associated Weyl group W, we show that the Chevalley morphism π : /W → /G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient → /G commutes, or does not commute, with base change on S.

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.

We show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.

We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

We describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

Let R and S be commutative rings, not necessarily with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of R×S. Unlike the case where R and S have an identity, an ideal (or primary ideal, or maximal ideal) of R×S need not be a ‘subproduct’ I×J of ideals. We show that for a ring R, for each commutative ring S every ideal (or primary ideal, or maximal ideal) is a subproduct if and only if R is an e-ring (that is, for r∈R, there exists er∈R with err=r) (or u-ring (that is, for each proper ideal A of R, )), the Abelian group (R/R2 ,+)has no maximal subgroups).

A proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, implies that either or . In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.

Let M and N be finitely generated and graded modules over a standard positive graded commutative Noetherian ring R, with irrelevant ideal R+. Let be the nth component of the graded generalized local cohomology module . In this paper we study the asymptotic behavior of Assf R+ () as n → –∞ whenever k is the least integer j for which the ordinary local cohomology module is not finitely generated.

For any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

In this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

This work is concerned with the question of when a von Neumann regular ring is expressible as a directed union of Artinian subrings.

Let R be an integral domain with quotient field K and let X be an indeterminate. A result of W. C. Waterhouse states that, if each quadratic polynomial f ∈ R[X] which factors into linear polynomials in K[X] also factors into linear polynomials in R[X], then every irreducible element in R is prime. In this note the rings which satisfy the hypothesis of this theorem are characterized, and compared to the rings for which each polynomial f ∈ R[X] which factors into two polynomials of positive degree in K[X] also factors into two polynomials of positive degree in R[X]. Relevant examples are furnished via the pullback construction.

Suppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

Let ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

An extension R1 of a right chain ring R is called immediate if R1 has the same residue division ring and the same lattice of principal right ideals as R. Properties of such immediate extensions are studied. It is proved that for every R, maximal immediate extensions exist, but that in contrast to the commutative case maximal right chain rings are not necessarily linearly compact.

For any ring S we define and describe its characteristic ring, k(S). It plays the rôle of the usual characteristic even in rings whose additive structure, (S, +), is complicated. The ring k(S) is an invariant of (S, +) and also reflects certain non-additive properties of S. If R is a left faithful ring without identity element, we show how to use k(R) to embed R in a ring R1 with identity. This unital overring of R inherits many ring properties of R; for instance, if R is artinian, noetherian, semiprime Goldie, regular, biregular or a V-ring, so too is R1. In the case of regularity (or generalizations thereof), R1 satisfies a universal property with respect to the adjunction of an identity