Using Gröbner Basis, we introduce a general algorithm to determine the additive structure of a module, when we know about it using indirect information about its structure. We apply the algorithm to determine the additive structure of indecomposable modules over ZCp, where Cp is the cyclic group of order a prime number p, and the p−pullback {Z→ Zp ← Z} of Z⊕Z.

Let Fq be the finite field with q elements, A := Fq[T ] and F := Fq(T ). Let φ be a Drinfeld A-module over F with trivial endomorphism ring. We prove analogues of the Erdös and Halberstam Theorems for φ. If φ has rank ≥ 3, we assume the validity of the Mumford-Tate Conjecture for φ. 1

We calculate the Grothendieck group K0(A), where A is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules algebras stable homotopy category SW CW-complexes. show that this direct sum free arising from localizations analogous to groups ideal classes maximal orders. As corollary, we obtain new simple proo...

We prove that the centralizer Cφ ⊆ HomR(M,M) of a nilpotent endomorphism φ of a finitely generated semisimple left R-module RM (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m×m matrix algebra Mm×m(R[t]), where m is the dimension (composition length) of ker(φ). If R is a finite dimensional division ring over its central subfield Z(R)...

Let D be an integral domain and E = {e1, . . . , ek} a finite nonempty subset of D. Then Int(E,D) has the strong two-generator property if and only if D is a Bezout domain. If D is a Dedekind domain which is not a principal ideal domain, then we characterize which elements of Int(E, D) are strong two-generators. Let D be an integral domain with quotient field K and E ⊆ D a subset of D. We let I...

Let K be a finite extension of the p-adic field Qp and let F (X,Y ) and G(X, Y ) be one-dimensional formal group laws over the ring of integers OK of K. Let φ(X) be a homomorphism from F to G which is defined over the ring of integers OCp of the completion Cp of Q p . In this paper we prove that if ker(φ) is finite then there is a discretely valued subfield L ⊂ Cp such that φ(X) is defined over...

The purpose of this paper is to address a number of issues raised by Avramov and Miller in a recent paper [1]. Let (R,m, k) be a Noetherian local ring of characteristic p > 0 with residue field k, and let φ : R → R be the the Frobenius homomorphism defined by φ(a) = a. For r ≥ 1, we denote by φrR the R-module structure on R via φ. That is, for a ∈ R and b ∈ φ r R, a · b = a r b. When R is a reg...

This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integ...

We present a constructive interpretation of some results in [6]. To any integral domain R we associate a distributive lattice V (R) which is a point-free presentation of the space of valuations over the ring R. Our definition, by generators and relations, is similar to Joyal’s definition of the Zariski lattice of a ring R, which is a point-free presentation of the Zariski spectrum of R. The spa...

For every action φ ∈ Hom(G, Autk(K)) of a group G on commutative ring K we introduce two abelian monoids. The monoid Cliffk(φ) consists equivalence classes strongly G-graded algebras type up to Clifford system extensions K-central algebras. $${{\cal C}_k}(\phi )$$ equivariance homomorphisms from the Picard groups (generalized collective characters). Furthermore, for such there is an exact seque...