From 07d4abc8a8eb5b1b9c9b8c2d5edc2e0c6613e98a Mon Sep 17 00:00:00 2001 From: Sixto Date: Thu, 22 May 2025 20:37:45 +0200 Subject: [PATCH] =?UTF-8?q?Aclaraci=C3=B3n=20de=20la=20noci=C3=B3n=20de=20?= =?UTF-8?q?divisibilidad=20entre=20monomios=20de=20un=20m=C3=B3dulo:=20aho?= =?UTF-8?q?ra,=20la=20definici=C3=B3n=20de=20divisibilidad=20se=20da=20en?= =?UTF-8?q?=20general=20para=20m=C3=B3dulos=20y,=20en=20base=20a=20esta,?= =?UTF-8?q?=20se=20explica=20para=20el=20caso=20de=20los=20monomios.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 194804 -> 194233 bytes TFG_Sixto_José_Martínez_Gan.tex | 16 ++++++++-------- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index ac4b7c7438343646d6a9634fd824c5941121e752..05ea5fbab77a114faf5dab39b22d0e5e1da2d1c5 100644 GIT binary patch delta 32929 zcmV(vK3 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a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -247,6 +247,9 @@ urlcolor=black]{hyperref} \end{equation*} En el caso que exista algún $G = \conjunto{g_1, g_2, \dots, g_n}$ de forma que $N = \engendrado{G}$ diremos que $N$ está finitamente generado y lo escribiremos como $N = \engendrado{g_1, g_2, \dots, g_n}$. \end{definicion} + \begin{definicion} \label{Definición Divisibilidad Módulos} + Sean $m$ y $l$ dos elementos de un \modulo{A} $M$. Diremos que $m$ divide a $l$ si existe $a \in A$ de forma que $am = l$. + \end{definicion} \begin{definicion} Sea $N$ un \submodulo{A} de $M$ y $\sim$ la siguiente relación de equivalencia sobre $M$: $m \sim m' \Leftrightarrow m - m' \in N$. Definimos el cociente de un módulo por un submódulo como $\cociente{M}{N} := \cociente{M}{\sim}$. Los elementos de este cociente se denotarán por $m + N$. \end{definicion} @@ -290,7 +293,7 @@ urlcolor=black]{hyperref} Diremos también que un \modulo{A} $M$ es noetheriano si todo submódulo $N \subseteq M$ está finitamente generado. Es decir, si existen $n_1, n_2, \dots, n_k$ en $N$ de forma que $N = \engendrado{n_1, n_2, \dots, n_k}_A$. \end{definicion} \begin{lema} - Sea $M$ un \modulo{A} noetheriano.\todo[inline]{¿A partir de qué momento se pueden suprimir las premisas genéricas?} Los submódulos y módulos cocientes de $M$ son también noetherianos. + Sea $M$ un \modulo{A} noetheriano. Los submódulos y módulos cocientes de $M$ son también noetherianos. \end{lema} \begin{proof} [\textsc{Demostración}] @@ -302,19 +305,19 @@ urlcolor=black]{hyperref} [\textsc{Demostración}] \end{proof} - \begin{definicion} + \begin{definicion} Sean $M$ y $N$ dos \modulo{A}s. El conjunto de homomorfismos de \modulo{A}s de $M$ a $N$ se denota por \begin{equation*} \Hom{A}{M}{N} := \definicionconjunto{\phi : M \longrightarrow N}{\text{$\phi$ es un homomorfismo de \modulo{A}s}}. \end{equation*} \end{definicion} - \begin{definicion} + \begin{definicion} Se define el módulo dual \dual{M} como el conjunto de homomorfismos de \modulo{A}s de $M$ en $A$: \begin{equation*} \dual{M} := \Hom{A}{M}{A}. \end{equation*} \end{definicion} - \begin{definicion} + \begin{definicion} \label{Definición Módulo Reflexivo} Sea $\psi : M \rightarrow \dualdual{M}$ el homomorfismo de \modulo{A}s que envía un elemento $m$ de $M$ a la función $\psi\!\left( m \right): \dual{M} \rightarrow A$ que a cada $\phi \in \dual{M}$ le asigna el valor $\phi\!\left( m \right)$: \begin{align*} \psi : M &\longrightarrow \Hom{A}{\Hom{A}{M}{A}}{A} \\ @@ -451,11 +454,8 @@ urlcolor=black]{hyperref} \end{itemize} \end{definicion} - Con este aparataje ya tenemos una noción de cuáles son los términos <> dentro de un polinomio, lo cual nos hace empezar a vislumbrar cómo dividir uno por otro. Por lo tanto, damos ya nuestro algoritmo de la división. + Con este aparataje ya tenemos una noción de cuáles son los términos <> dentro de un polinomio, lo cual nos hace empezar a vislumbrar cómo dividir uno por otro. Lo primero es entender cuándo pueden dividirse dos monomios de $\Kx^r$. Según la definición \ref{Definición Divisibilidad Módulos}, para $x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $x^\mathbf{\beta}\vectorbasecanonica{j}$ dos monomios de \Kx{} tenemos que $x^\mathbf{\alpha}\vectorbasecanonica{i}$ divide a $x^\mathbf{\beta}\vectorbasecanonica{j}$ si $i = j$ y $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. Ahora sí, podemos dar un algoritmo de la división. - \begin{definicion} - Sean $x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $x^\mathbf{\beta}\vectorbasecanonica{j}$ dos monomios de \Kx. Diremos que $x^\mathbf{\alpha}\vectorbasecanonica{i}$ divide a $x^\mathbf{\beta}\vectorbasecanonica{j}$ y lo notaremos por $x^\mathbf{\alpha}\vectorbasecanonica{i} \vert x^\mathbf{\beta}\vectorbasecanonica{j}$ si $i = j$ y $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. - \end{definicion} \begin{definicion}[algoritmo de la división] Supongamos que tenemos dos polinomios $f$ y $g$ en \Kx. Para dividir $f$ entre $g$ aplicaremos un algoritmo que consiste en replicar el algoritmo que ya conocemos para los polinomios en una variable. Sin embargo, ahora los términos que guían los coeficientes que vamos poniendo son $\LT{f}$ y $\LT{g}$: