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In §1 we define some properties of ideals by using games. These properties strengthen precipitousness. We call these stronger ideals. In §2 we show some limitations on the existence of such ideals over ${}_{\kappa }\lambda$. We also present a consistency result concerning the existence of such ideals over ${}_{\kappa }\lambda$. In §3 we show that such ideals satisfy stronger normality. We show a cardinal arithmetical consequence of the existence of strongly normal ideals. In § 4 we study some “large cardinal-like” consequences of stronger...