We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.

There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize three ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including hereditary, residual, and cofunctorial, and see how they interact with other properties such as the finitistic property.

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

In this paper, we explore a generalization of the notion of integrality. In particular, we study a near-integrality condition that is intermediate between the concepts of integral and almost integral. This property (referred to as the $\Omega $-almost integral property) is a representative independent specialization of the standard notion of almost integrality. Some of the properties of this generalization are explored in this paper, and these properties are compared with the notion of pseudo-integrality introduced by Anderson, Houston, and Zafrullah. Additionally, it is shown that the $\Omega $-almost integral property serves to characterize the survival/lying over pairs of Dobbs and Coykendall.

It is proved that if u is an element of a faithful algebra over a commutative ring R, then u satisfies a polynomial over R which has unit content if and only if the extension R ⊂ R[u] has the imcomparability property. Applications include new proofs of results of Gilmer-Hoffmann and Papick, as well as a characterization of the P-extensions introduced by Gilmer and Hoffmann.