2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67326We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $χ$-conditions of Artin and Zhang and the strong noetherian property have very different behavior on the left and right sides.17 Pages, revised version: significant changes--introduction rewritten, new section on tensor products added, main theorem restated at endRings and Algebras16W50; 14A22text