Bi-atomic classes of positive semirings

Semigroup Forum - John Meakin has had a distinguished career of over half a century in the theory of semigroups. This article gives a synopsis of his most important contributions. The author, a...     

Cut Vertices in Zero-Divisor Graphs of Finite Commutative Rings

A cut vertex of a connected graph is a vertex whose removal would result in a graph having two or more connected components. We examine the presence of cut vertices in zero-divisor graphs of finite commutative rings and provide a partial classification of the rings in which they appear.     

Polar and singular value decomposition of 3×3 magic squares

In the literature there have been reported a number of studies of probabilities involved in Monopoly, the worldwide popular game designed by Charles B. Darrow in the 1930s. Simple assumptions often lead to inaccurate results. In this paper, having distinguished between ‘In Jail’ and ‘Just Visiting’, both theoretical computations and computer simulations are conducted to determine the probability of landing on ‘JAIL’. Theoretical and simulation results are consistent. The sum of the probabilities of landing on ‘In Jail’ and ‘Just Visiting’ is found to be 6.40%, both in theory and in simulation, compared with Ian Stewart's 5.89%. The long-term probabilities of landing on the other 39 spaces are computed and it is concluded that the best properties to buy are the orange ones.     

A fundamental theorem of modular arithmetic

Periodica Mathematica Hungarica - The Fundamental Theorem of Arithmetic is a statement about the uniqueness of factorization in the ring of integers. The notation and proof easily generalize to...     

Number theory of matrix semigroups

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M_n(Z), the semigroup of n×n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.     

A Krull-Schmidt theorem for one-dimensional rings of finite Cohen-Macaulay type

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.     

Bounds for indecomposable torsion-free modules

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M_P over each field R_P (R localized at P), where P ranges over the minimal prime ideals of R. We assume that there exists a bound N_R on the ranks of all indecomposable finitely generated torsion-free R-modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is (r,r,…,r), then r∈{1,2,3,4,6}, even when R is not local.