Summary: Given a locally \(m\)-convex algebra, we examine when a quasi-singular element of it becomes a topological divisor of zero, in an appropriate sense, to which one is forced here, for lack of unity in the given algebra. This is achieved for quasi-singular elements of type (A) and (B) in certain locally \(m\)-convex algebras. Furthermore, in the context of suitable locally \(m\)-convex algebras, topological zero divisors are roots of Šilov points, while the latter characterize also non-removable ideals. On the other hand, a characterization of Šilov points, accordingly of non-removable ideals, is obtained by means of topological divisors of zero, realized by the same net. Finally, we also characterize advertibly complete locally \(m\)-convex algebras through topological divisors of zero.