Local Gorenstein ring such that the square of the maximal ideal contains every minimal prime ideal

Question

Let $(R, \mathfrak m)$ be a commutative local Gorenstein ring such that every minimal prime ideal of $R$ is contained in $\mathfrak m^2$. Then is it true that $R$ is an integral domain ?

If this is not true in general, what if we also assume $R$ is either reduced (has no non-zero nilpotent) or irreducible (has exactly one minimal prime) ?

Answer 1

None of what you say may be true. Take $R=\mathbb{C}[[x,y,z]]/(x^2+y^2+z^2)^2$. Then it is irreducible, the only minimal prime is generated by $x^2+y^2+z^2$, whichis contained in the square of the maximal ideal and it is Gorenstein.

Similarly, take $R=\mathbb{C}[[x_1,x_2,x_3, y_1,y_2, y_3]]/ (\sum x_i^2)(\sum y_i^2)$. It is reduced and has all the other properties you want.