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In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial…

In this paper, we study properties of nodal orders defined over arbitrary base fields. In particular we give a classification of complete real nodal orders.

The present paper studies structure of the ring of integer-valued entire functions. We characterize certain classes of prime and maximal ideals and investigate some of their properties.

We classify quantum cellular automata whose cells are qubits, on hypercubic lattices $\mathbb Z^s$, with the von Neumann neighborhood scheme, in terms of realizability as finite-depth quantum circuits. We show the most general structure of…

Let $T$ be a subset of a ring $A$, and let $M$ be an $A$-module. We study the additive subgroups $F$ of $M$ such that, for all $x \in M$, if $tx \in F$ for some $t \in T$, then $x \in F$. We call any such subset $F$ of $M$ a $T$-factroid of…

In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings $A$ in…

We completely determine all semigroup varieties satysfiyng a permutational identity of length 3 that are modular elements of the lattice of all semigroup varieties. Using this result, we provide an example of a semigroup variety that is a…

For the cyclic group $C_2$ we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring $\underline{\mathbb{Z}/\ell}$, for $\ell$ a prime. This is fairly simple for $\ell$ odd, but for…

Let $p$ be a prime, $W$ the ring of Witt vectors of a perfect field $k$ of characteristic $p$ and $\zeta$ a primitive $p$th root of unity. We introduce a new notion of calculus over $W$ that we call absolute calculus. It may be seen as a…

We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same…

An $R$-module $M$ is called absolutely self pure if for any finitely generated left ideal of $R$ whose kernel is in the filter generated by the set of all left ideals $L$ of $R$ with $L \supseteq$ ann $(m)$ for some $m \in M$, any map from…

A complete classification is given for one dimensional chains with nearest neighbor interactions having two states in each site, for which a matrix product ground state exists. The Hamiltonians and their corresponding matrix product ground…

Let $M$ be a module. A {\em $\delta$-cover} of $M$ is an epimorphism from a module $F$ onto $M$ with a $\delta$-small kernel. A $\delta$-cover is said to be a {\em flat $\delta$-cover} in case $F$ is a flat module. In the present paper, we…

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For…

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if…

Fully commutative elements in types $B$ and $D$ are completely characterized and counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended the study of fully commutative elements from Coxeter groups to the complex setting, giving…

Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $. $M$ is said to be a content $R$-module if $x \in c(x)M $, for all…

Suppose O is a complete discrete valuation ring of positive characteristic with perfect residue field. The category of finite flat strict modules was introduced recently by Faltings and appears as an equal characteristic analogue of the…

We give a new proof of the fact that any finite quadratic module can be decomposed into indecomposable ones. For any indecomposable finite quadratic module, we construct a lattice, and a positive definite lattice, both of which are of the…

We consider the category of linear relations over an arbitrary commutative ring, and identify it as a subcategory of the category of Kronecker representations. We observe that this subcategory forms a definable, faithful and hereditary…