Characteristic-free test ideals

Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 p>0 , in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.

Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

Theory of g-Tg-Exterior and g-Tg-Frontier Operators: Definitions, Essential Properties and, Consistent, Independent Axioms

In a generalized topological space Tg = (Ω, Tg), generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, are merely two of a number of generalized primitive operators which may be employed to topologize the underlying set Ω in the generalized sense. Generalized exterior and generalized frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, are other generalized primitive operators by means of which characterizations of generalized operations under g-Intg, g-Clg : P (Ω) −→ P (Ω) can be given without even realizing generalized interior and generalized closure operations first in order to topologize Ω in the generalized sense. In a recent work, the present authors have defined novel types of generalized interior and generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg and studied their essential properties and commutativity. In this work, they propose to present novel definitions of generalized exterior and generalized frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, a set of consistent, independent axioms after studying their essential properties, and established further characterizations of generalized operations under g-Intg, g-Clg : P (Ω) −→ P (Ω) in Tg.