Consider the ideal $I$ in $\mathbb{C}[z_{1},\ldots,z_{m}]$ generated by $\{p_{1},\ldots,p_{t}\}$, where $t\leq m$ and $p_{1},\ldots,p_{t}$ intersects completely i.e. the map $(p_{1},\ldots,p_{t}):\mathbb{C}^{m}\mapsto\mathbb{C}^{t}$ is a submersion $\forall w\in Z(I)$. If furthermore, we assume that $Z(I)$ is connected, then is it true that $I$ is prime?
I am not sure how complete intersection influences the condition. For example, if we relax the condition $Z(I)$ being connected and consider the ideal $\langle z_{1}z_{2},z_{1}+z_{2}-1\rangle \subset \mathbb{C}[z_{1},z_{2}],$ then $z_{1},z_{2}\notin I$, which means $I$ is not prime. But I am unable to find counterexamples that fits both the conditions and still not a prime ideal.
Also, note that complete intersection forces $\{p_{1},\ldots,p_{t}\}$ to be a minimal generator of $I$.