Example of reduced ring which is not integral domain

$\begingroup$

So, a reduced ring has no nonzero nilpotent elements, while integral domain has no nonzero zero divisors. Of course, the latter is way stronger then the first condition, implying that every integral domain needs to be reduced ring. But, having $(\exists n \geq 2) x^n = 0 \Longleftrightarrow x = 0$ does not up to my knowledge imply $xy=0 \Longleftrightarrow (x = 0 \vee y = 0)$.

That is why I need example of reduced ring which is not integral domain. Thanks!

$\endgroup$ 2

1 Answer

$\begingroup$

$\Bbb Z[X,Y]/(XY)$ has no non-zero nilpotents, but $XY = 0$.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy