Multiplicative closure of generalized invertible elements in a ring

Let $R$ be a ring. It is shown that the subset $R^{\{1\}}$ of $\{1\}$-invertible elements of $R$ is a multiplicative set if and only if $R$ is an SSP ring. As a quick application, it is proved that the upper triangular matrix ring ${\rm T}_n(R)$ $(n\geq 2)$ over any ring $R$ is never SSP. In the case of a ring $R$ with involution, multiplicative property of $R^{\{1,3\}}$, $R^{\{1,4\}}$ and $R^{\dagger}$ are also characterized. It is shown that $R^{\{1,3\}}$ is a multiplicative set if and only if $R^{\{1,4\}}$ is a multiplicative set, which also implies that $R^{\dagger}$ is a multiplicative set.