We call a theory a Dedekind theory if every complete quantifier-free type with one free variable either has a trivial positive part or it is isolated by a positive quantifier-free formula. The theory of vector spaces and the theory fields are examples. We prove that in a Dedekind theory all positive quantifier-free types are principal so, in a sense, Dedekind theories are Noetherian. We show th...

(c) μ(rs,m) = μ(r, μ(s,m)) (d) if 1 ∈ R, then μ(1,m) = m. We shall usually omit the notation of μ and simply write r ·m for μ(r,m). Thus axiom (c) would be written (rs) ·m = r · (s ·m), etc. Exercise 1. We denote by EndGrp(M) the set of group endomorphisms of M : An element φ ∈ EndGrp(M) is a group homomorphism φ : M → M . EndGrp(M) is naturally a ring, with addition and multiplication defined ...

In 2011, Khurana, Lam and Wang defined the following property: (*) A commutative unital ring satisfies property “power stable range one” if for all a,b∈A with aA+bA=A there is an integer N=N(a,b)≥1 λ=λ(a,b)∈A such that bN+λa∈A×, unit group of A. 2019, Berman Erman considered rings (**) has enough homogeneous polynomials any k≥1 set S:={p1,p2,…⁡,pk}, primitive points in An n≥2, exists a polynomi...

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational poly-topes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer...

This research aims to give the decompositions of a finitely generated module over some special ring, such as principal ideal domain and Dedekind domain. One main problems with theory is analyze objects module. was using literature study on modules topics from scientific journals, especially those related theory. And by selective cases we find pattern build conjecture or hypothesis, deductive pr...

Consider the space R∆ of rational functions of r variables with poles on an arrangement of hyperplanes ∆. It is important to study the decomposition of the space R∆ under the action of the ring of differential operators with constant coefficients. In the one variable case, a rational function of z with poles at most on z = 0 is written uniquely as φ(z) = Princ(φ)(z)+ψ(z) where Princ(φ)(z) = ∑ n...

Let R be a commutative ring with identity. Let φ : I(R) → I(R) ∪ {∅} be a function where I(R) denotes the set of all ideals of R. A proper ideal Q of R is called φ-primary if whenever a, b ∈ R, ab ∈ Q−φ(Q) implies that either a ∈ Q or b ∈ √ Q. So if we take φ∅(Q) = ∅ (resp., φ0(Q) = 0), a φ-primary ideal is primary (resp., weakly primary). In this paper we study the properties of several genera...

A finite module M over a noetherian local ring R is said to be Gorenstein if Ext(k, M) = 0 for all i 6= dimR. A endomorphism φ : R → R of rings is called contracting if φ(m) ⊆ m for some i ≥ 1. Letting R denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR( R,M) ∼= M and ExtiR( R,M) = 0 for 1 ≤ i ≤ depthR.

We prove a necessary and sufficient criterion for the ring of integer-valued polynomials to behave well under localization. Then, we study how Picard group Int(D) quotient P ( D ) : = Pic Int / $\mathcal {P}(D):=\mathrm{Pic}(\mathrm{Int}(D))/\mathrm{Pic}(D)$ in relation Jaffard, weak pre-Jaffard families; particular, show that ≃ ⨁ T {P}(D)\simeq \bigoplus \mathcal {P}(T)$ when ranges Jaffard fa...