6 problems

Let $\Sigma$ be a ring and let $\alpha,\beta\in\operatorname{Sper}\Sigma$. The connectedness property at $\alpha$ and $\beta$ requires that, for every finite collection…

Let $\Sigma$ be a ring, let $f\in\Sigma$ be nonzero, and let $\alpha,\beta\in\operatorname{Sper}\Sigma$ have a common specialization. The strong separation property for…

Let $\Sigma$ be a ring. The strong connectedness property is required at pairs $\alpha,\beta\in\operatorname{Sper}\Sigma$ having a common specialization, with the defining connecte…

Strong Connectedness conjecture. A regular ring $\Sigma$ has the Strong Connectedness Property at every pair of points $\alpha,\beta\in\operatorname{Sper}\Sigma$ having a common sp…

Definable connectedness conjecture. A regular ring $A$ satisfies the definable connectedness property at every pair $\alpha,\beta\in\operatorname{Sper} A$.

Let $A$ be a regular ring, let $\alpha,\beta\in\operatorname{Sper} A$, and assume that either $A$ is complete or $\sqrt{\langle\alpha,\beta\rangle}=\mathfrak m$. Let…