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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, a tertiary ideal is an (two-sided) ideal in a (perhaps noncommutative) ring that cannot be expressed as a nontrivial intersection of a right fractional ideal with another ideal. Tertiary ideals generalize primary ideals to the case of noncommutative rings. Although primary decompositions do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian. Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutatitve rings.