Frobenius templates in certain $2 \times 2$ matrix rings

TL;DR

This paper investigates Frobenius problems within specific rings of upper triangular 2x2 matrices with constant diagonal, extending classical number theory concepts to matrix rings.

Contribution

It introduces Frobenius problems in matrix rings of upper triangular 2x2 matrices with constant diagonal, expanding the scope of classical Frobenius problem research.

Findings

Extended Frobenius problem to matrix rings

Derived solutions for specific matrix ring structures

Connected matrix Frobenius problems to classical integer cases

Abstract

The classical Frobenius problem is to find the largest integer that cannot be written as a linear combination of a given set of positive, coprime integers using nonnegative integer coefficients. Prior work has generalized the classical Frobenius problem from integers to Frobenius problems in other rings. This paper explores Frobenius problems in various rings of (upper) triangular $2 \times 2$ matrices with constant diagonal.