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A commutative noetherian local ring      ( R , m ) (R,\mathfrak {m})       is Gorenstein if and only if every parameter ideal of      R R       is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer      ℓ  \ell       (depending on      R R      ) such that      R R       is Gorenstein if and only if there exists an irreducible parameter ideal contained in      m ℓ  \mathfrak {m}^\ell      . We give upper bounds for      ℓ  \ell       that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.

Reducibility of parameter ideals in low powers of the maximal ideal