I currently have, the following two commutative diagrams for the second and third ring isomorphism theorems:
Second Isomorphism
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes,fit}
\begin{document}
\begin{tikzpicture}[x=1.75cm,y=1.75cm]
\node (r) at (0,2) {$R$};
\node (apb) at (0,1) {$A+B$};
\node (a) at (-1,0) {$A$};
\node (b) at (1,0) {$B$};
\node(ab) at (0,-1) {$A\cap B$};
\draw (r)--(apb)--(a)--(ab) (ab)--(b)--(apb);
\node[rotate=45,ellipse,draw,dashed,inner xsep=-9mm,inner ysep=1mm,fit=(apb)(b)] {};
\node[rotate=45,ellipse,draw,dashed,inner xsep=-9mm,inner ysep=1mm,fit=(ab)(a)] {};
\node {$\cong$};
\end{tikzpicture}
\end{document}
Third Isomorphism
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes,fit}
\begin{document}
\begin{tikzpicture}[x=1.75cm,y=1.75cm]
\node (r) at (-1,1) {$R$};
\node (j) at (-1,0) {$J$};
\node (i) at (-1,-1) {$I$};
\node (ri) at (1,1) {$R/I$};
\node (ji) at (1,0) {$J/I$};
\node (o) at (1,-1) {$0$};
\draw (r)--(j)--(i) (ri)--(ji)--(o);
\node[rotate=0,ellipse,draw,dashed,inner xsep=1mm,inner ysep=-2mm,fit=(r)(j)] {};
\node[rotate=0,ellipse,draw,dashed,inner xsep=1mm,inner ysep=-2mm,fit=(ri)(ji)] {};
\end{tikzpicture}
\end{document}
- First off, is there anyway to make these using
tikz-cd
package include the encircling ellipses. For the first diagram, I had a tikzcd picture but I wasn't able to find a way to add the ellipses. (I prefer tikz-cd because it is much more straightforward when creating these diagrams). - As you can see, I wasn't able to complete the diagram for the Third Isomorphism, because I'm trying to get it so the arrow between the two encircled ellipses at heigh between the first and second nodes, at the level of their connecting lines. Can someone help me with that? (I also want a "isomorphic" i.e $\cong$ symbol as the caption of the connecting arrow).
Thanks!