Abstract The quasi-ideals of a structural matrix ring, which are quasi-ideals solely by virtue of the shape of the matrices they comprise, and their idealizers are described. The structural subrings of a fixed structural matrix ring that can be written as idealizers of structural quasi-ideals of the ring are characterized and the relationship with idealizers of structural one-sided ideals is pointed out. It is shown that properties of structural matrix rings can sometimes be used to rouse an expectation of how rings in general will behave with respect to those properties. The similarity and difference between a known result, stating that the semiprimeness of a ring R is a sufficient but not necessary condition for every minimal quasi-ideal of R to be the intersection of a minimal left and a minimal right ideal of R, and a similar result in structural matrix rings, are discussed. An example is given of a matrix ring over an arbitrary division ring for which the existence of a minimal quasi-ideal, the exist...