We consider the problem of commutative rank computation of a given matrix space, B ⊆ Fn×n. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is n, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient det...

The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative rin...

Totalistic cellular automata were introduced by Wolfram [1, 2] as a means of studying higher-color cellular automata within a reasonable space of possible rules. The underlying principle is to systematically construct a subset of rules within all the possible rules of the highercolor cellular automata, and the system in totalistic cellular automata is to add up the values of the cells within th...

7 We consider polynomial-time Turing machines that have access to two oracles and investigate when the order of oracle queries is significant. The oracles used here are complete languages for the Polynomial Hierarchy (PH). We prove that, for solving decision problems, the order of oracle queries does not matter. This improves upon the previous result of Hemaspaandra, Hemaspaandra and Hempel, wh...

Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2...

A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to commutative rings, prove that commutative n-SRC rings (n ≥ 2) are precisely the commutative local ring...

Around 1980 Connes extended the notions of geometry to the noncommutative setting. Since then non-commutative geometry has turned into a very active area of mathematical research. As a first non-trivial example of a noncommutative manifold Connes discussed subalgebras of rotation algebras, the socalled non-commutative tori. In the last two decades researchers have unrevealed the relevance of no...

The critical properties of the real φ scalar field theory are studied numerically on the fuzzy sphere. The fuzzy sphere is a finite matrix (non–commutative) approximation of the algebra of functions on the usual two dimensional sphere. It is also one of the simplest examples of a non–commutative space to study field theory on. Aside from the usual disordered and uniform phases present in the co...

In this paper, we investigate the behavior of non-commutative IR divergences and will also discuss their cancellation in the physical cross sections. The commutative IR (soft) divergences existing in the non-planar diagrams will be examined in order to prove an all order cancellation of these divergences using the Weinberg’s method. In noncommutative QED, collinear divergences due to triple pho...

The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...