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Abstract
An ideal I in a (commutative unitary) ring A is strongly irreducible if, for all ideals J, K ◃ A, the inclusion J ∩ K ⊆ I implies J ⊆ I or K ⊆ I. Trivial examples are prime ideals. The set of strongly irreducible ideals is viewed as a subspace of the spectral space of all ideals. To a large extent, the study of strongly irreducible ideals can be reduced from arbitrary rings to local rings. It is shown how one can recognize whether an ideal in a local ring is strongly irreducible. The existence of a nonprime strongly irreducible ideal has a strong impact on the structure of a local ring. Then the ring carries a truncated valuation (a notion generalizing classical valuations), which defines a totally ordered set of valuation ideals. Many valuation ideals are also strongly irreducible.