A finitely presented module is a concept from abstract algebra, specifically within the theory of modules over rings. It refers to a module that can be described using a finite number of generators and relations. Such modules are of particular interest because they can be studied using computational tools and provide insights into more general algebraic structures.
In terms of usage frequency, it is more common in written academic contexts, especially in mathematics and computer science literature, rather than in oral speech. The term is utilized primarily by mathematicians and students in higher education.
Example Sentences:
1. "A finitely presented module allows mathematicians to utilize computational group theory for structural analysis."
"Un módulo presentado de forma finita permite a los matemáticos utilizar la teoría de grupos computacionales para el análisis estructural."
"In the study of algebraic topology, finitely presented modules play a pivotal role in understanding homology."
"En el estudio de la topología algebraica, los módulos presentados de forma finita juegan un papel clave en la comprensión de la homología."
"To determine whether a finitely presented module is free entails a deep understanding of its generators and relations."
"Determinar si un módulo presentado de forma finita es libre implica un profundo entendimiento de sus generadores y relaciones."
The term "finitely presented module" does not have specific idiomatic expressions associated with it, largely due to its technical nature. However, we can discuss some common phrases and examples related to the concept of modules within mathematical contexts where "finitely presented" might appear in more extended discussions.
Example Sentences with Related Concepts:
1. "In computational algebra, many results can be formulated in the setting of finitely presented modules, leading to efficient algorithms."
"En álgebra computacional, muchos resultados pueden ser formulados en el contexto de módulos presentados de forma finita, lo que lleva a algoritmos eficientes."
"The classification of finitely presented modules can drastically affect the structure of the underlying ring."
"La clasificación de módulos presentados de forma finita puede afectar drásticamente la estructura del anillo subyacente."
"When dealing with finitely presented modules, one often needs to apply homological algebra techniques."
"Al tratar con módulos presentados de forma finita, a menudo es necesario aplicar técnicas de álgebra homológica."
The term "finitely presented module" can be broken down into its components: - Finitely derives from the Latin word "finitus," meaning "limited or confined." - Presented comes from the Latin "praesentare," meaning "to present or show." - Module stems from the Latin "modulus," which means "a small measure" or "standard."
The phrase has evolved to encapsulate the notion of a module represented in a limited, manageable way through generators and relations in mathematical discourse.
This information encapsulates the concept of "finitely presented module" within mathematical contexts, emphasizing its significance and application in the field of abstract algebra.