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\Materia{Álgebra II}
\Cuatrimestre{Primer Cuatrimestre --- 2016}
\Practica{6}
\Titulo{Módulos - Segunda parte}

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\begin{document}

\maketitle

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\section{Módulos noetherianos y artinianos}
%%%%%%%%%%
\begin{ejercicios}

%%%%%%%%%%
\item Demostrar las siguientes afirmaciones.
\begin{ejitems}
\item Un $\k$-espacio vectorial $V$ es noetheriano si y solo si $\dim_\k V < \infty$.
\item Una $\k$-álgebra de dimensión finita es noetheriana. ¿Vale la vuelta?
\end{ejitems}

%%%%%%%%%%
\item Probar que un anillo en el que todo ideal a izquierda es principal es noetheriano a izquierda.


%%%%%%%%%%
\item Sean $A$ un anillo, $M$ un $A$-módulo a izquierda y $f\in\End_A(M)$.
Para cada $n\in\NN_0$ definimos $K_n=\ker f^n$ e $I_n=\im f^n$. Probar que:
\begin{ejitems}
\item si $K_1=K_2$ entonces $K_1\cap I_1=0$;
\item si $I_1=I_2$ entonces $K_1+I_1=M$;
\item si $M$ es noetheriano existe $n\in\NN_0$ tal que $K_n\cap I_n=0$;
\item si $M$ es noetheriano y $f$ es sobreyectivo entonces $f$ es un
automorfismo.
\end{ejitems}

%%%%%%%%%%
\item Sea $d\in\ZZ$ y sea $\sqrt d\in\CC$ una raíz cuadrada de $d$. Probar que
$\ZZ[\sqrt d]$ es un anillo noetheriano.

\item\header{Anillos de matrices} Sea $n\in\NN$. Probar que un anillo $A$ es
  noetheriano a izquierda, resp. a derecha, si y solo si $M_n(A)$ es
  noetheriano a izquierda, resp. a derecha.

%%%%%%%%%%

\item Probar que un dominio íntegro artiniano es un cuerpo.

%%%%%%%%%%
\item Probar las siguientes afirmaciones.
\begin{ejitems}
\item Un grupo abeliano artiniano es de torsión.
\item Un grupo abeliano es artiniano y noetheriano si y solo si es finito.
\end{ejitems}

%%%%%%%%%%
\item\header{Extensiones finitas de anillos}  Sean $A$ un anillo y $B \subseteq A$
  un subanillo tal que $A$ es finitamente generado como $B$-módulo a izquierda.
  Probar que si $B$ es noetheriano a izquierda, entonces $A$ es noetheriano a
  izquierda.

%%%%%%%%%%
\item\header{Álgebras de matrices formales} Sean $A$ y $B$ anillos y sea $M$ un
$A$-$B$-bimódulo. Sea $T$ el grupo abeliano $A \oplus M \oplus B$. Dados $a \in
A, m \in M$ y $b \in B$, notamos por $\begin{psmallmatrix} a & m \\ 0 &
b\end{psmallmatrix}$ al elemento $(a,m,b) \in T$.
\begin{ejitems}
\item Probar que la fórmula
\begin{align*}
  \begin{pmatrix} 
    a & m \\ 0 & b
  \end{pmatrix} \cdot
  \begin{pmatrix} 
    a' & m' \\ 0 & b'
  \end{pmatrix}
= \begin{pmatrix} 
    aa' & am' + mb' \\ 0 & bb'
  \end{pmatrix}
\end{align*}
define un producto asociativo en $T$. Concluir que $(T,+,\cdot)$ es un anillo,
el cual también notamos por $\begin{psmallmatrix} A & M \\ 0 & B
\end{psmallmatrix}$.

\item Probar que $T$ es noetheriano a izquierda, resp. a derecha, si y solo si
  $A$ y $B$ son noetherianos a izquierda, resp. a derecha, y $M$ es finitamente
  generado como $A$-módulo a izquierda, resp. $B$-módulo a  derecha. 

\item Probar que $T = \begin{psmallmatrix} \ZZ & \QQ \\ 0 & \QQ
\end{psmallmatrix}$ no es isomorfo a su anillo opuesto.
\end{ejitems}


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\end{ejercicios}
\end{document}
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