1
$\begingroup$

Let $A$ be a commutative ring with unit, and $P$ a prime ideal.

My question is:

If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample?

Of course, if $(I+P)/P$ is reducible in $A/P$, then $I+P$ is reducible in $A$, because $$ \dfrac{I+P}{P} = \dfrac{J_1}{P} + \dfrac{J_2}{P} \Rightarrow I+P = J_1 + J_2 $$ but i don't think this is enough to conclude that $I$ is reducible.

$\endgroup$

1 Answer 1

1
$\begingroup$

Here is a counterexample:

$I=(xy-z)$ is irreducible (it is even prime) in $K[x,y,z]$, but $(I+(z))/(z)=(xy)$ is not irreducible in $K[x,y,z]/(z)=K[x,y]$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .