In the most widely accepted definition of matrix near-rings [\textit{J. D. P. Meldrum} and \textit{A. P. J. van der Walt}, Arch. Math. 47, 312-319 (1986; Zbl 0611.16025)], there are two obvious ways of linking ideals in the base near-ring to ideals in the matrix near-ring. While these two ideals coincide in matrix rings, they can differ widely in matrix near-rings [\textit{J. D. P. Meldrum} and \textit{J. H. Meyer}, Commun. Algebra 24, No. 5, 1601-1619 (1996; Zbl 0859.16041)]. The first author has generalized the concept of a matrix near-ring [Commun. Algebra 24, No. 6, 2065-2077 (1996; Zbl 0856.16042)]. In this paper ideals in centralizer near-rings are studied, and this is then applied to generalized matrix near-rings over centralizer near-rings. The main result shows that if every ideal of the centralizer near-ring is an annihilator ideal, then, as for rings, the correspondence between ideals in the base near-ring and those in the matrix near-ring is bijective. Such centralizer near-rings are shown to exist, indeed to be quite numerous.