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Anillos. Propiedades de los anillos. Subanillos. Tipos de anillos. Característica. Divisores de cero. Homomorfismos e isomorfismos. Ideales. Ideales principales. Ideales primos y maximales. Anillos cocientes. Anillos euclidianos.
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Date
2021-12-17
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Universidad Nacional de Educación Enrique Guzmán y Valle
Abstract
El objetivo de este trabajo de investigaciòn es dar a conocer que un anillo es un conjunto no vacío en el que se encuentran definidas dos
operaciones binarias internas, llamadas suma o adición, denotada por +, y producto o
multiplicación, denotada por ∙, tal que se cumple que el anillo definido en la suma es un
grupo abeliano, que el anillo definido en la multiplicación es un semigrupo que cuenta con
el elemento de identidad 1 y que la operación de multiplicación es distributiva con
respecto a la suma para todos sus elementos. Estos anillos pueden tener diversas
características y, por lo tanto, pueden ser clasificados de acuerdo con la relación que existe
entre todas estas. Entre los tipos de anillos se encuentran los anillos conmutativos, los
anillos con identidad, los anillos de dominio integral, los anillos de división y los cuerpos.
En un anillo conmutativo se cumple la propiedad de conmutatividad de la multiplicación.
Un anillo con identidad es aquel en el que se cumple que existe un elemento 1 que
pertenece al conjunto. Decimos que un anillo es un dominio de integridad si este es un
anillo conmutativo con identidad en el que se cumple que a ∙ b = 0 solo si a = 0 o b = 0.
Por otra parte, un anillo de división es un anillo no nulo con identidad en el que todos sus
elementos que no son iguales a cero son invertibles y, por lo tanto, son unidad. Finalmente,
un cuerpo es aquel anillo de división que, además, es conmutativo. Todos estos presentan
características distintas, pero se encuentran relacionados al compartir algunas propiedades.
Cualquiera de estos, al encontrarse definidos por las operaciones internas de adición y
multiplicación pueden contener dentro de sí mismos otros anillos más pequeños, los cuales
se denominan subanillos. Estos subanillos se caracterizan por tener las mismas operaciones
internas que definen al anillo original.
Dentro del campo de los anillos existen funciones que preservan las leyes de
operación interna existentes entre estas. Estas se denominan homomorfismos, y se
caracterizan por ser funciones 𝜙������:𝐵������ → 𝐶������ que satisface: 𝜙������(𝑏������ + 𝑐������) = 𝜙������(𝑏������) + 𝜙������(𝑐������) y 𝜙������(𝑏������ ∙ 𝑐������) =
𝜙������(𝑏������) ∙ 𝜙������(𝑐������). De acuerdo con sus características, los homomorfismos pueden ser
clasificadas en monomorfismo (si es una función inyectiva), epimorfismo (si es una
función suprayectiva) e isomorfismo ( si es una función biyectiva, es decir, inyectiva y
suprayectiva). Estas funciones son importantes porque permiten establecer la propiedades
de las operaciones que se realizan entre distintos anillos.
Dentro de la la teoría de anillos, el ideal es una de las subestructuras más
importantes. Esta importancia radica en que este concepto permite la construcción de
anillos cociente y porque, además, es uno de los puntos más importantes para el desarrollo
de otras áreas, tales como la teoría de números y la gemetría algebraica. Un ideal es un
subanillo que cumpre dos premisas: (a) es un subgrupo aditivo del anillo original y (b) para
todos los elementos del subanillo y el anillo se cumple que la multiplicación entre estos
también son elementos del subanillo. Así como existen tipos de anillos, también se pueden
definir tipos de ideales. Entre estos se encuentran los ideales generados, ideal principal,
ideal primo e ideal maximal. Tras definir el concepto de ideal, se puede dar paso al anillo
cociente. El anillo cociente es el anillo definido por un subanillo A de un anillo R (R/A),
en el que se cumple (1) (s+A)+(p+A)=s+p+A y (2) (s+A)∙(p+A)=s∙p+A
The objective of this research work is to show that a ring is a non-empty set in which two internal binary operations, called addition or addition, denoted by +, and product or multiplication, denoted by ∙, such that the ring defined in the sum is a abelian group, that the ring defined in the multiplication is a semigroup that has the identity element 1 and that the multiplication operation is distributive with with respect to the sum for all its elements. These rings can have different characteristics and, therefore, can be classified according to the relationship that exists among all these Among the types of rings are commutative rings, rings with identity, integral domain rings, split rings, and fields. In a commutative ring, the commutative property of multiplication holds. A ring with identity is one in which it is fulfilled that there is an element 1 that belongs to the set. We say that a ring is an integrity domain if it is a commutative ring with identity in which a ∙ b = 0 holds only if a = 0 or b = 0. On the other hand, a division ring is a nonzero ring with identity in which all its Elements that are not equal to zero are invertible and therefore are units. Finally, a field is that ring of division that, moreover, is commutative. All of these present distinct characteristics, but are related by sharing some properties. Any of these, being defined by the internal operations of addition and multiplication can contain within themselves other smaller rings, which They are called subrings. These subrings are characterized by having the same operations internal parts that define the original ring. Within the field of rings there are functions that preserve the laws of internal operation existing between them. These are called homomorphisms, and are characterized by being functions 𝜙�����:𝐵����� → 𝐶����� that satisfies: 𝜙�����(𝑏����� + 𝑐�����) = 𝜙�����(𝑏�����) + 𝜙�����(𝑐�����) and 𝜙�����(𝑏����� ∙ 𝑐�����) = 𝜙�����(𝑏�����) ∙ 𝜙�����(𝑐�����). According to their characteristics, homomorphisms can be classified in monomorphism (if it is an injective function), epimorphism (if it is a surjective function) and isomorphism (if it is a bijective function, that is, injective and surjective). These functions are important because they allow you to set the properties of the operations carried out between different rings. Within ring theory, the ideal is one of the most important substructures. important. This importance lies in the fact that this concept allows the construction of quotient rings and because, in addition, it is one of the most important points for the development from other areas, such as number theory and algebraic geometry. An ideal is a subring that satisfies two premises: (a) it is an additive subgroup of the original ring and (b) for all the elements of the subring and the ring it is fulfilled that the multiplication between these they are also elements of the subring. Just as there are types of rings, you can also define types of ideals. Among these are the generated ideals, main ideal, prime ideal and maximal ideal. After defining the concept of ideal, we can give way to the ring quotient. The quotient ring is the ring defined by a subring A of a ring R (R/A), in which (1) (s+A)+(p+A)=s+p+A and (2) (s+A)∙(p+A)=s∙p+A holds
The objective of this research work is to show that a ring is a non-empty set in which two internal binary operations, called addition or addition, denoted by +, and product or multiplication, denoted by ∙, such that the ring defined in the sum is a abelian group, that the ring defined in the multiplication is a semigroup that has the identity element 1 and that the multiplication operation is distributive with with respect to the sum for all its elements. These rings can have different characteristics and, therefore, can be classified according to the relationship that exists among all these Among the types of rings are commutative rings, rings with identity, integral domain rings, split rings, and fields. In a commutative ring, the commutative property of multiplication holds. A ring with identity is one in which it is fulfilled that there is an element 1 that belongs to the set. We say that a ring is an integrity domain if it is a commutative ring with identity in which a ∙ b = 0 holds only if a = 0 or b = 0. On the other hand, a division ring is a nonzero ring with identity in which all its Elements that are not equal to zero are invertible and therefore are units. Finally, a field is that ring of division that, moreover, is commutative. All of these present distinct characteristics, but are related by sharing some properties. Any of these, being defined by the internal operations of addition and multiplication can contain within themselves other smaller rings, which They are called subrings. These subrings are characterized by having the same operations internal parts that define the original ring. Within the field of rings there are functions that preserve the laws of internal operation existing between them. These are called homomorphisms, and are characterized by being functions 𝜙�����:𝐵����� → 𝐶����� that satisfies: 𝜙�����(𝑏����� + 𝑐�����) = 𝜙�����(𝑏�����) + 𝜙�����(𝑐�����) and 𝜙�����(𝑏����� ∙ 𝑐�����) = 𝜙�����(𝑏�����) ∙ 𝜙�����(𝑐�����). According to their characteristics, homomorphisms can be classified in monomorphism (if it is an injective function), epimorphism (if it is a surjective function) and isomorphism (if it is a bijective function, that is, injective and surjective). These functions are important because they allow you to set the properties of the operations carried out between different rings. Within ring theory, the ideal is one of the most important substructures. important. This importance lies in the fact that this concept allows the construction of quotient rings and because, in addition, it is one of the most important points for the development from other areas, such as number theory and algebraic geometry. An ideal is a subring that satisfies two premises: (a) it is an additive subgroup of the original ring and (b) for all the elements of the subring and the ring it is fulfilled that the multiplication between these they are also elements of the subring. Just as there are types of rings, you can also define types of ideals. Among these are the generated ideals, main ideal, prime ideal and maximal ideal. After defining the concept of ideal, we can give way to the ring quotient. The quotient ring is the ring defined by a subring A of a ring R (R/A), in which (1) (s+A)+(p+A)=s+p+A and (2) (s+A)∙(p+A)=s∙p+A holds
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Citation
Chujutalli Mori T. (2021). Anillos. Propiedades de los anillos. Subanillos. Tipos de anillos.
Característica. Divisores de cero. Homomorfismos e isomorfismos.
Ideales. Ideales principales. Ideales primos y maximales. Anillos
cocientes. Anillos euclidianos (Monografía de pregrado). Universidad Nacional de Educación Enrique Guzmán y Valle, Lima, Perú.