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The problem is:

Show that every group and ring is congruence-permutable, but not necessarily congruence-distributive.

I know that in group every normal sub group has permutable property and in every ring, every ideal has permutable property. Moreover, for not being distributive I should give an example. please Guide me.


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What do "congruence-permutable" and "congruence-distributive" mean? – Chris Eagle Jan 25 '13 at 10:44

If you compute the entire lattice of normal subgroups of the Klein 4-group, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, you will see you get the smallest modular non-distributive lattice, $M_3$.

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Let $K$ be a field and $K[x,y]$ the ring of polynomials on $x$ and $y$, and coefficients in $K$. Let $I$ be the ideal of $K[x,y]$ generated by the polynomial $f(x,y)=x$; let $J$ be the ideal generated by $g(x,y)=y$; and $H$ be the ideal generated by $h(x,y)=x+y$.

It's easy to verify that $(I \vee J) \wedge K \nleq (I \wedge K) \vee (J \wedge K)$, so that $K[x,y]$ is not congruence-distributive. EDIT: an example of a polynomial testifying this is $k(x,y)=x+y$,

The congruence lattice of $K[x,y]$ is a sub-lattice of the congruence lattice of its group reduct; thus you also get a group which is not congruence-distributive.

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