A broad theory of factorization: from matrices to ideals

A number of problems in various areas of mathematics, commonly known as factorization problems, revolve around the feasibility or impossibility of decomposing specific objects (referred to as elements by mathematicians) into others that are simpler and, in a sense, cannot be further broken down into smaller pieces. These simpler objects are typically called "irreducibles" and serve as the fundamental components of factorizations (another term for decompositions). Let`s consider a familiar example: positive integers. In this case, prime numbers are the irreducible components. Prime numbers are those integers that cannot be divided into smaller positive integers greater than one, and every positive integer can be expressed as a product of prime numbers. Similarly, polynomials with integer, real, or complex coefficients can be factored into irreducible polynomials. Positive integers have a unique factorization into prime numbers, but for many other algebraic objects, there are numerous ways to break them down into irreducible building blocks. Factorization theory studies the existence and non-uniqueness (however defined) of factorizations of algebraic objects into irreducible ones. The objective is to describe all the distinct factorizations of a fixed element, using suitable algebraic parameters such as sets of (factorization) lengths. If an element x is a product of n irreducibles, then n is a factorization length of x. The study of factorizations of given objects, in fact, provides a better understanding of their nature and structure. The project, "A broad theory of factorization: from matrices to ideals," investigates algebraic structures, including rings of matrices and monoids of ideals, from the perspective of factorization theory. These structures have not been explored from this viewpoint before, and the new approach will shed fresh light on long-standing open problems. Additionally, it will lead to a significant extension of current methods for investigating non-unique factorizations in rings and monoids, offering diverse and innovative applications.