# Tertiary ideal

An ideal $I$ of a ring $R$ which cannot be expressed as the intersection of a right fractional ideal $r(I,A)$ and an ideal $B$, each strictly larger than $I$. All irreducible ideals are tertiary. In Noetherian rings, tertiary ideals are the same as primary ideals (cf. Additive theory of ideals; Primary decomposition).
Suppose that the ring $R$ satisfies the maximum condition for left and right fractional ideals, and that every ideal decomposes as an intersection of finitely many indecomposable ideals. Then for every ideal $Q$ there exists a tertiary radical, $\mathrm{ter}(Q)$, the largest ideal $T$ of $R$ such that, for any ideal $B$, $$r(Q,T) \cap B = Q \ \Rightarrow\ B=Q \ .$$
An analysis of the properties of left and right fractions (of ideals of a ring, of submodules of a module, and others) leads to systems with fractions in which the general notions of $S$-primarity and $S$-primary radicals occur naturally. This allows one to formulate the "intersection" , "existence" and "uniqueness theorems" as axioms. In this approach, tertiarity is the unique notion of primarity for which all these three theorems hold, i.e. it is the unique "good" generalization of classical primarity (cf. , ).