# Primary decomposition of the zero ideal when the nilradical is prime

Let $$A$$ be a commutative ring and $$N$$ its nilradical. Assume $$N$$ is a prime ideal of $$A$$. Then it is known that Spec $$A$$ is irreducible for the Zariski topology. However the zero ideal may be not primary.

Is there a concrete example of this situation and what is the primary decomposition of the zero ideal? Does it decompose Spec $$A$$, even though it is irreducible?

• @math54321 thanks, if you make this comment into an answer I'll accept it. By the way x^2 and xy are quotiented out, so no need to put them in the second primary ideal. May 14 at 17:48
• That's true, but writing it as $(x^2, xy, y^n)$ makes it obviously primary to $(x,y)$ May 14 at 17:55

This happens precisely when $$A$$ has embedded primes. The classic example is $$A = k[x,y]/(x^2,xy)$$: here the nilradical is $$(x)$$, and there are infinitely many primary decompositions of $$0$$ in $$A$$, arising from $$(x^2,xy) = (x) \cap (x^2,xy,y^n)$$ in $$k[x,y]$$ for any $$n \ge 2$$.