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Diremos que $M$ es \emph{libre de torsión} si $\mathrm{Tors}\!\left( M \right) = 0$.\todo{Un comentario sobre que todo espacio vectorial es libre, que en módulos libre implica libre de torsión y que la recíproca no es cierta.} + Diremos que $M$ es \emph{libre de torsión} si $\mathrm{Tors}\!\left( M \right) = 0$. \end{definicion} \begin{ejemplo} Pensemos en \enteros{} y en \enterosmodulo{p} donde $p \in \naturales$ como en \modulo{\enteros}. El módulo \enteros{} admite a $1$ como base ya que todo entero $z$ tiene una única representación como $z\cdot 1$. Ahora bien, supongamos que \enterosmodulo{p} tiene una base \conjunto{z_1, z_2, \dots, z_k}. Para cualquier $z \in \enterosmodulo{p}$ existirán $a_1, a_2, \dots, a_n$ de forma que $z = \sum_{i = 1}^{k}a_iz_i$. Ahora bien, démonos cuenta de que \enterosmodulo{p} no es un \modulo{\enteros} libre de torsión ya que para todo $z$, $pz = \sum_{i = 1}^{k}pa_iz_i = 0$; esto nos permite escribir @@ -403,6 +403,8 @@ urlcolor=black]{hyperref} \end{equation*} Por lo tanto, la representación de $z$ no es única en términos de la base y, por ende, \enterosmodulo{p} no es libre. \end{ejemplo} + De hecho, todo módulo libre es libre de torsión, aunque el converso no tiene por qué ser cierto. Un caso donde este converso es falso es el \nolbreaks{Ejemplo \ref{Ejemplo de No Tener Base}}. + Por último, damos las dos nociones más básicas relacionadas con la divisibilidad en los módulos. \begin{definicion} \label{Definición Divisibilidad Módulos} Sean $m$ y $l$ dos elementos de un \modulo{A} $M$. Diremos que $m$ \emph{divide a} $l$ si existe $a \in A$ de forma que $am = l$. @@ -1240,7 +1242,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 En el capítulo anterior concluíamos con un método para calcular \derivaciones{f} utilizando las sicigias, una herramienta que hemos derivado de las bases de Gröbner. Con estas, podemos obtener un conjunto generador de \derivaciones{f} e incluso una base de Gröbner de este mismo que nos permita determinar si una derivación está o no en el conjunto. Pero otra pregunta de gran importancia aparece cuando consideramos una gran diferencia entre módulos y espacios vectoriales: así como para estos últimos siempre tenemos que los subespacios admiten una base, en el caso de los módulos no tiene por qué ser así. - \begin{ejemplo} + \begin{ejemplo} \label{Ejemplo de No Tener Base} Consideremos el \modulo{\anillodepolinomios{K}{x, y}} \engendrado{x, y}. Por ser noetheriano, sabemos que, de existir alguna base para \engendrado{x, y}, esta debe de ser finita. Supongamos que existe y es $\conjunto{a_1, a_2, \dots, a_k}$. Entonces, existirán unos únicos $p_1, p_2, \dots, p_k$ y $q_1, q_2, \dots, q_k$ en \anillodepolinomios{K}{x, y} de forma que \begin{equation*} p_1a_1 + p_2a_2 + \dots + p_ka_k = x \quad \text{y} \quad q_1a_1 + q_2a_2 + \dots + q_ka_k = y.