Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose removal disconnects the graph, in both compressed and non-compressed zero-divisor graphs. In doing so, we connect these graph-theoretic concepts with algebraic notions and provide realization theorems of zero-divisor graphs for commutative rings with identity.

We review facts about rank, multilinear rank, multiplex rank and generic rank of tensors as well as folding of a tensor into a matrix of multihomogeneous polynomials. We define the new concept of folding rank of tensors and compare its properties to other ranks. We review the concept of determinantal schemes associated to a tensor. Then we define the new concept of a folding generic tensor meaning that all its determinantal schemes behave generically. Our main theorem states that for ``small'' 3-tensors, any folding generic tensor has generic rank, and the reverse does not always hold.

In dimension 2, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means of finding their defining equations.

We introduce the concept of an $s$-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an $s$-Hankel hypermatrix: the ideal $\I {s}{t}$ generated by certain $2 \times 2$ slice minors, and the ideal $\It {s}{t}$ generated by certain $2 \times 2$ generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of $\I {s}{t}$, and provide the explicit list of minimal primes for large values of $s$. Finally we give some geometrical interpretations and generalize a theorem of Watanabe.

In this paper, we build off of Takahashi and White's $\catpc $-projective dimension and $\catic $-injective dimension to define these dimensions for when $C$ is a semidaulizing complex. We develop the framework for these homological dimensions by establishing base change results and local-global behavior. Furthermore, we investigate how these dimensions interact with other invariants.