Let $R$ be a ring and $P(t)\in R[t]$ be a polynomial. An element $c\in R$ is called a pseudo-root of $P$ if $P(t)=Q(t)(c-t)L(t)$ where $Q,L\in R[t]$. We introduce and study a universal algebra of such pseudo-roots and show its relations to "noncommutative logic" and "noncommutative topology"