Locally direct sums [W, Definition 3.15] appeared naturally in classification results for topological rings (see, e.g.,[K2], [S1], [S2], [S3], [U1]). We give here a result (Theorem 3) for locally compact Baer rings by using of locally direct sums. 1. Conventions and definitions All topological rings are assumed associative and Hausdorff. The subring generated by a subset A of a ring R is denote...

The classical ring of integer-valued polynomials Int(Z) consists of the polynomials in Q[X] that map Z into Z. We consider a generalization of integervalued polynomials where elements of Q[X] act on sets such as rings of algebraic integers or the ring of n× n matrices with entries in Z. The collection of polynomials thus produced is a subring of Int(Z), and the principal question we consider is...

We propose a polynomial time f -algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over Fq) for computing an isomorphism (if there is any) of a finite dimensional Fq(x)-algebra A given by structure constants with the algebra of n by n matrices with entries from Fq(x). The method is based on computing a finite Fq-subalgebra of A which is the intersecti...

Let IA be the toric ideal of a homogeneous normal configuration A ⊂ Z . We prove that IA is generated by circuits if and only if each unbalanced circuit of IA has a “connector” which is a linear combination of circuits with a square-free term. In particular if each circuit of IA with non-square-free terms is balanced, then IA is generated by circuits. As a consequence we prove that the toric id...

problem: Given: A field K and n variables x1, . . . , xn and m polynomials yi = pi(x1, . . . , xn) ∈ K[x1, . . . , xn] for i = 1, . . . ,m. (1) Aim: Find a presentation for the subring K[y] := K[y1, . . . , ym] of K[x] := K[x1, . . . , xn]. Invariants: The difference of m and the transcendence degree of K(y) := K(y1, . . . , ym) over K will be called the deficiency d = d(y) of the tuple y in K(...

Abstract. We give a polynomial-time algorithm of computing the classical Hurwitz numbers Hg,d, which were defined by Hurwitz 125 years ago. We show that the generating series of Hg,d for any fixed g > 2 lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the ...

Hilbert was concerned with fundamental problems of invariant theory: given a linear group, G, acting linearly on the ring of polynomials, S = K[X1, . . . , XN ], we let S be the subring of invariants. Is S nitely generated as an algebra over the eld, K, and if so, what are its generators? Assuming it is nitely generated, that is, that S = K[Y1, . . . , YN ′ ]/I, is it the case that I is nitely ...

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is ExtR(G,G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe the...

Notation 1.1. Let X be a projective variety over an algebraically closed field. For every integer k ≥ 0, denote by Nk(X) the finitely-generated free Abelian group of k-cycles modulo numerical equivalence, and denote by N(X) the k graded piece of the quotient algebra A∗(X)/Num∗(X), cf. [Ful98, Example 19.3.9]. For every Z-module B, denote Nk(X)B := Nk(X)⊗B, resp. N(X)B := N(X)⊗B. Denote by NEk(X...

Closed string tachyon condensation has been studied in orbifolds C/ZN,p of flat space, using the chiral ring of the underlying N = 2 conformal field theory. Here we show that similar phenomena occur in the curved smooth background obtained by adding NS5-branes, such that chiral tachyons are localised on lens submanifolds SU(2)/ZN,p. We find a level-independent subring which coincides with that ...