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.
To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as Then t(I) always contains I.
If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals