I am wondering right now whether every ring, which is commutative, has a 1, and is reduced (i.e. no nilpotent elements) always has just one minimal prime Ideal.
Does anyone have a proof or a counterexample? The motivation for this question is that said rings correspond to coordinate rings $k[X]$ for irreducible algebraic sets $X$. And the minimal prime ideals should correspond to the irreducible components of $X$, which in case that $X$ itself is irreducible should just be $X$. Possible it is necessary to assume that $k$ is also an integral domain.
As there was some confusion, because I did not phrase the question clearly (sorry for that) and the answer is so simple, here is the short solution: If $R$ is an integral domain then $0$ is a prime ideal, and clearly it is also the unique minimal prime ideal. If $R=k[X]$ (i.e. $R$ is the coordinate ring of $X$), then $0$ corresponds to $X$ and $X$ is irreducible.