Let D be a division ring and let m,n be integers ≥ 2. Let Mm×n(D) be the space of m × n matrices. In the fundamental theorem of the geometry of rectangular matrices all bijective mappings φ of Mm×n(D) are determined such that both φ and φ−1 preserve adjacency. We show that if a bijective map φ of Mm×n(D) preserves the adjacency then also φ −1 preserves the adjacency. Thus the supposition that φ...

There are many instances of the principle that if A is an algebra of functions on X, then every ring homomorphism A → C is given by evaluation at a particular x ∈ X. Examples are the nullstellensatz in algebraic geometry and the result of Gelfand when X is a compact Hausdorff space. In these cases one can regard X as being included in Hom(A, C) as the set of those f : A → C which satisfy the se...

The Briançon-Skoda theorem appears in many variations in recent literature. The common denominator is that the theorem gives a sufficient condition that implies a membership φ ∈ a, where a is an ideal of some ring R. In the analytic interpretation R is the local ring of an analytic space Z, and the condition is that |φ| ≤ C|a| holds on the space Z. The theorem thus relates the rate of vanishing...

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 ...

A self stabilizing distributed system is a network of state machines which, regardless of its initial global state, once started will regain its consistency by itself in a nite number of steps. Two are the main issues in the design of a self-stabilizing system: stabilization time and memory requirements. In this paper we present an almost two state self stabilizing unidirectional system solving...

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...

In the cohomology ring of an extraspecial p-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers. Introduction Methods to determine the cohomology ring of a finite group almost always presuppose that...

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...