Consider the ring $R$ of non commutative real polynomials in $X$ and $Y$. Denote by $I$ the principal two-sided ideal generated by $X$ and $J$ the principal two-sided ideal generated by $XY+1$. Then $I+J=R$, but $I∩J≠IJ$.
I understand the first part since $1\in I+J$, but I could not show $I∩J≠IJ$. Any example to show? Thanks
Chinese remainder theorem does not hold in non commutative case
idealsring-theory
Best Answer
The key idea here is that if the ring doesn't commute, we don't necessarily have $IJ=JI$ whereas $I \cap J= J \cap I$. So consider the element $(XY+1)\cdot X$. It is in $I \cap J$ but not in $IJ$ hence $I \cap J \neq IJ$.