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In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove thateach ideal from such class satisfies the conclusion of the conjecture in question. We also study the relation between the class of Burch ideals and that of the ideals we define, and construct several examples that corroborate our results., This is a preprint of an article published in Collectanea Mathematica. The final authenticated version is available online at: https://doi.org/10.1007/s13348-021-00315-8

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring \((R,\mathfrak m)\) is Gorenstein if and only if it admits an integrally closed \(\mathfrak m\)-primary ideal of finite Gorenstein dimension. This is accomplished through a detailed study of certain test complexes. Along the way we construct such a test complex that detect finiteness of Gorenstein dimension, but not that of projective dimension.