## Notes

## Notas sobre módulos y anillos

Por varios años he estado escribiendo unas notas sobre teoría de anillos y módulos, como un intento de libro de texto para estudiantes de licenciatura. Estas notas están basadas en los cursos que tomé con el Prof. Alejandro Alvarado en la Facultad de Ciencias, UNAM cuando era estudiante. Mi idea con estas notas, es dar un libro de texto en español con las nociones básicas de la teoría de anillos y módulos. Actualmente tengo dos versiones de estas notas:

- La primera contiene todo el material para un curso de un año y los contenidos están organizados como yo tomé los cursos. A estas notas las he llamado "Una introducción a la teoría de anillos y módulos"
- La segunda es un arreglo de las primeras que hice para dar el curso de semestre de Algebra Moderna 3 en la primavera del 2023. Estas segundas notas contienen máso menos la mitad de material que las primeras, pero en cada capítulo se da una lista de ejercicios pensados para los alumnos de licenciatura. A estas notas las he llamado "Una introducción a la teoría de módulos sobre anillos no conmutativos. Vol. 1"

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The first one contains all the material for a one year course in the order I took the lectures as student. "Una introducción a la teoría de anillos y módulos"

The second one is an arrangement of the previous notes as I gave the lectures in Spring 2023. This second notes comprehend, so only one semester course, so it contains just the half of the complete notes, but in this shorter notes each chapter has a list of exercises and many mistake have been corrected. "Una introducción a la teoría de módulos sobre anillos no comutativos. Vol. 1"

It has to be said that either of these versions can be considered finished.

*English*) For many years I have been typing a textbook on rings and modules for undergraduate students. This notes are based on the lectures gave by Prof. Alejandro Alvarado at Facultad de Ciencias, UNAM when I was student. My idea was to give to the students a textbook, written in spanish, to start in the theory of rings and modules. Right now I have two versions of these notes:The first one contains all the material for a one year course in the order I took the lectures as student. "Una introducción a la teoría de anillos y módulos"

The second one is an arrangement of the previous notes as I gave the lectures in Spring 2023. This second notes comprehend, so only one semester course, so it contains just the half of the complete notes, but in this shorter notes each chapter has a list of exercises and many mistake have been corrected. "Una introducción a la teoría de módulos sobre anillos no comutativos. Vol. 1"

It has to be said that either of these versions can be considered finished.

## Other notes

- As a Postdoc researcher at Benemérita Universidad Autónoma de Puebla (BUAP), I was requested by a student to gave a graduated course in which we could read the paper "Tilting modules and tilting torsion theories" written by R. Colpi and J. Trlifaj. At the end I typed my notes of the course. We started with some basic homological algebra and then we filled all the gaps to understand the paper. Along the course some exercises were left to my student. At the end we got these notes.

- In Northern Illinois University, in joint with Prof. John Beachy, we started a seminar to read the book "Lectures on algebraic quantum groups" written by K. Brown and K. Goodearl. In order to understand the book and give the presentations at the seminar, I typed the basic concepts and many computations to fill the gaps in the book. These notes do not cover all the book and they are incomplete.

## Some code

I have been interested in programming for a while, in a basic level. I started to learn python in order to try to compute examples for my research. In particular, I was interested in to compute ideals and submodules of finite rings and finite modules and check some properties. Also, I wanted to compute some functions on finite lattices and check some of their properties. Since these computations were not the goal in my research, my code is typed in a very basic level. Perhaps there is a library which can make it easy but I wanted to practice my skills. Below I give some of the small programs I have made.

- In this file I put together the files to compute the ideals of a finite ring, with some minor modifications, these files can be used to compute submodules. First we have to give the set of elements of the ring and define the multiplication and addition in this set. After that the code will compute the ideals, it will display how many they are and the elements of each one. Also, the code can compute prime ideals, zero-divisors and regular elements.

Ideals of a ring - At some point of my research, I was trying to find an indecomposable no uniform \(R\)-module \(M\). Since I had tested my code in some different rings, I was able to find this example.

Indecomposable no uniform module - Some of my research deals with lattices and some "point-free techniques". There are some important functions on a lattice called derivatives and some important derivatives called nucleus. In a frame there is always an implication and this implication determines other two nucleus called v and w (usually). These nucleus are important because with them it is possible to compute all the nucleus. In connection with my work "Some operators and dimensions in modular meet-continuous lattices", I wrote a code to compute the totalizers in a finite lattice. In the following file you can find the code to compute these derivatives and the Cantor-Bendixon derivative.

Operators - Since 2021, I started to learn about linear morphisms on a lattice (introduced by Toma Albu). So, I wanted to develop a code to compute the linear endomorphisms on a lattice and check some of their properties. In the following file you can find the code to do this with some examples. Also, a part of the code is to plot the Hasse diagram of the lattice. This code is related with my papers "\(\mathfrak{m}\)-Baer and \(\mathfrak{m}\)-Rickart lattices" and "\(\mathfrak{m}\)-endoregular lattices".

Linear endomorphisms