In this paper we give a generalization of the notion—introduced by C. Traverso in [5]— of “glueing of prime ideals of a ring $B$ over a prime ideal of a ring $A \subset B$”. We prove the following result: if $B$ is an overring of $A$, integral of finite type over $A$, $A$ can be obtained from $B$ by a finite number of glueings of the above defined kind; geometrically, every singularity of an algebraic variety $V$ can be obtained from the normalization of $V$, by a finite number of such glueings.