Rings and polynomials

A compactness property of the k-abelian monoids

We study the unification problem for a class of equational theories that comprises important examples like abelian monoids (AC), idempotent abelian monoids (ACI), and abelian groups. Monoidal theories have the common characteristic that unification algorithms are based on solving linear equation systems over a semiring.The close correspondence between unification and linear algebra can be used to characterize the unification type of monoidal theories in purely algebraic terms, and an application of Hilbert's Basis Theorem gives a sufficient criterion for a monoidal theory to be unitary.

Noetherian Semigroup Rings with Several Objects

The surprising fact that Hilbert's basis theorem in algebra shows identifiabilty of ideals of polynomials in the limit from positive data is derived by the correspondence between ideals and languages in the context of machine learning. This correspondence also reveals the difference between the two and raises new problems to be solved in both of algebra and machine learning. In this article we solve the problem of providing a concrete form of the characteristic set of a union of two polynomial ideals. Our previous work showed that the finite basis of every polynomial ideal is its characteristic set, which ensures that the class of ideals of polynomials is identifiable from positive data. Union or settheoretic sum is a basic set operation, and it could be conjectured that there is some effective method which produces a characteristic set of a union of two polynomial ideals if both of the basis of ideals are given. Unfortunately, we cannot find a previous work which gives a general method for how to find characteristic sets of unions of languages even though the languages are in a class identifiable from positive data. We give methods for computing a characteristic set of the union of two polynomial ideals.

Weakly multiplicative coactions of quantized function algebras

Let [Formula: see text] be a finitely generated associative algebra over a field of characteristic different from 2. Herstein asked when the Lie algebra [Formula: see text] is finitely generated. Recently, it was shown that for a finitely generated nil algebra [Formula: see text] all derived powers of [Formula: see text] are finitely generated Lie algebras. Let [Formula: see text] be the Lie algebra of skew-symmetric elements of an associative algebra with involution. We consider all derived powers of the Lie algebra [Formula: see text] and prove that for any finitely generated associative nil algebra with an involution, all derived powers of [Formula: see text] are finitely generated Lie algebras.

A non-zero element of the Lie algebra \({\mathfrak{se}(3)}\) of the special Euclidean spatial isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev (Publ. Res. Inst. Math. Sci. 4:1199–1257, 2007) gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where \({k\geq3}\). Following Study, Geometrie der Dynamen, (1903), we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants.

For a simply-connected closed manifold X of $\dim X \neq 4$ , the mapping class group $\pi _0(\mathrm {Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each $k>0$ , we prove that there are simply-connected closed smooth 4-manifolds X for which $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ are not finitely generated. The infinitely generated subgroup of $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ which we detect are topologically trivial, and unstable under the connected sum of $S^2 \times S^2$ . These results are proven by constructing and computing an infinite family of characteristic classes using Seiberg–Witten theory.

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

Let PR be a projective module with trace ideal T. An R-module XR is T-accessible if XT = X. If PR is finitely generated projective and C is the R-endomorphism ring of PR, such that CPR, then for XR, Hom (PR, XR)C is artinian (noetherian) if and only if XR satisfies the minimum (maximum) condition on T-accessible submodules. Further, if XR is T-accessible then Hom (PR, XR)C is finitely generated if and only if XR is finitely generated. The purpose of the present paper is to investigate cHom (CPR, XR), where PR is a finitely generated projective R-module and C = End (PR), the R-endomorphism ring of P, with respect to the properties of chain conditions and finite generation. Throughout this paper R is a ring with identity and all modules over R are unitary. The convention of writing module-homomorphisms on the side opposite the scalars is adopted here. 1. Preliminaries. Let PR be a finitely generated projective R-module with C = End (PR) such that CPR. The dual module of PR is (with respect to RR), RPC = Hom (CPR, RRR). It is well known, see [1], that the map PR 6P PR* = Hom (RP*, RR) given by p p, where ffp = fp, forf E P* is an R-isomorphism. LEMMA 1.1. For PR finitely generated projective, C = End (PR), the map End (PR) -End (RP*) given by c -c j, where fc= fc is a ring isomorphism. PROOF. The above map is nothing more than the composite of the following maps Hom (1, 6p) Hom (PR, PR) RHom (PR pR*)

We compute the finite generation ideal for Daigle and Freudenburg’s counterexample to Hilbert’s fourteenth problem. This ideal helps to understand how far the ring of invariants is from being finitely generated. Our calculations show that the finite generation ideal is the radical of an ideal generated by three infinite families of invariants. We show that these three families together with an additional invariant form a SAGBI-basis of the ring of invariants. We use the properties of our SAGBI-basis in our computation of the finite generation ideal.

Given a generically finite local extension of valuation rings , the question of whether W is the localization of a finitely generated V‐algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper, we prove Knaf's conjecture in full generality.

Finite generation and presentability of wreath products of monoids

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.