In this paper we consider the class $\mathcal{P}_1(R)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when $\mathcal{P}_1(R)$ is a covering class. More precisely, we investigate Enochs' Conjecture for this class, that is the question of whether $\mathcal{P}_1(R)$ is covering necessarily implies that $\mathcal{P}_1(R)$ is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring $R$. This gives an example of a cotorsion pair $(\mathcal{P}_1(R), \mathcal{P}_1(R)^\perp)$ which is not necessarily of finite type such that $\mathcal{P}_1(R)$ satisfies Enochs' Conjecture. Moreover, we describe the class $\varinjlim \mathcal{P}_1(R)$ over (not-necessarily commutative) rings which admit a classical ring of quotients.
Let $\mathcal{R}_e$ be a finite commutative chain ring with nilpotency index $e \geq 2.$ In this paper, all repeated-root constacyclic codes of arbitrary lengths over $\mathcal{R}_{2},$ their sizes and their dual codes are determined. As an application, some isodual constacyclic codes over $\mathcal{R}_{2}$ are also listed. Moreover, Hamming distances, Rosenbloom-Tsfasman distances and Rosenbloom-Tsfasman weight distributions of all repeated-root constacyclic codes over $\mathcal{R}_{2}$ and some repeated-root constacyclic codes over $\mathcal{R}_{e}$ are determined.
We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{C}^{\infty}-$rings instead of ordinary commutative unital rings, looking for similar results to those one finds in the latter, and expanding some others presented in [20]. We give an explicit description of an adjunction between the categories $\mathcal{C}^\infty{\rm Rng}$ and ${\rm CRing}$, in order to study this "bridge". We present and prove many properties of the analog of the radical of an ideal of a ring (nam…
Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated to the $1$-tilting class $\mathcal{T}$ over a commutative ring as illustrated by Hrbek. We prove that a $1$-tilting class $\mathcal{T}$ is enveloping if and only if $ \mathcal{G}$ is a perfect Gabriel topology (that is, it arises from a perfect localisation) and $R/J$ is a perfect ring for each $J \in \mathcal{G}$, or equivalently $\mathcal{G}$ is a perfect G…
We are interested in characterising the commutative rings for which a $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$ provides for covers, that is when the class $\mathcal{A}$ is a covering class. We use Hrbek's bijective correspondence between the $1$-tilting cotorsion pairs over a commutative ring $R$ and the faithful finitely generated Gabriel topologies on $R$. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for $1$-tilting cotorsion pairs arising from flat injective ring epimorphis…
Let $R$ be a commutative ring with nonzero identity, and $δ:\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ the set of all ideals of $R$. In this paper, we introduce the concept of $δ$-$n$-ideals which is an extension of $n$-ideals in commutative rings. We call a proper ideal $I$ of $R$ a $δ$-$n$-ideal if whenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in δ(I)$. For example, $δ_{1}$ is defined by $δ_{1}(I)=\sqrt{I}.$ A number of results and characterizations related to $δ$-$n$-ideals are given. Furthermore, we present some results relate…
Let $\mathcal{I}(R)$ be the set of all ideals of a ring $R$, $δ$ be an expansion function of $\mathcal{I}(R)$. In this paper, the $δ$-$J$-ideal of a commutative ring is defined, that is, if $a, b\in R$ and $ab\in I\in \mathcal{I}(R)$, then $a\in J(R)$ (the Jacobson radical of $R$) or $b\in δ(I)$. Moreover, some properties of $δ$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.
In this paper we state and prove ad hoc "Separation Theorems" of the so-called Smooth Commutative Algebra, the Commutative Algebra of \(\mathcal{C}^{\infty}-\)rings. These results are formally similar to the ones we find in (ordinary) Commutative Algebra. However, their proof is not so straightforward, since it depends on the introduction of the concept of "smooth saturation". As an application of these theorems we present an interesting result that sheds light on the connections between the smooth Zariski spectrum and the real smooth spectrum of a \(\mathcal{C}^{\infty}-\)ring.
Let $H$ be an atomic monoid. For $x \in H$, let $\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\mathcal{L}(H) = \{\mathsf{L}(x) \mid x \in H\}$. On the other hand, the elasticity of $x$, denoted by $ρ(x)$, is the quotient $\sup \mathsf{L}(x)/\inf \mathsf{L}(x)$ and the elasticity of $H$ is the supremum of the set $\{ρ(x) \mid x \in H\}$. The system of sets of lengths and the elasticity of $H$ both measure how far is $H$ from being half-factorial, i.e., $|\mathsf{L}(x)| = 1$ for each $x \in H$. …
Let $\mathcal{S}$ be a finite commutative semigroup written additively, and let $\exp(\mathcal{S})$ be its exponent which is defined as the least common multiple of all periods of the elements in $\mathcal{S}$. For every sequence $T$ of elements in $\mathcal{S}$ (repetition allowed), let $σ(T) \in \mathcal{S}$ denote the sum of all terms of $T$. Define the Davenport constant $D(\mathcal{S})$ of $\mathcal{S}$ to be the least positive integer $d$ such that every sequence $T$ over $\mathcal{S}$ of length at least $d$ contains a proper subsequence $T'$ with $σ(T')=σ(T)$, and define the Erdős-Ginzb…
Let $\mathcal{S}$ be a commutative semigroup, and let $T$ be a sequence of terms from the semigroup $\mathcal{S}$. We call $T$ an (additively) {\sl irreducible} sequence provided that no sum of its some terms vanishes. Given any element $a$ of $\mathcal{S}$, let ${\rm D}_a(\mathcal{S})$ be the largest length of the irreducible sequence such that the sum of all terms from the sequence is equal to $a$. In case that any ascending chain of principal ideals starting from the ideal $(a)$ terminates in $\mathcal{S}$, we found the sufficient and necessary conditions of ${\rm D}_a(\mathcal{S})$ being f…
If $\mathbb{F}$ is an ordered field and $M$ is a finite-rank torsion-free monoid, then one can embed $M$ into a finite-dimensional vector space over $\mathbb{F}$ via the inclusion $M \hookrightarrow \text{gp}(M) \hookrightarrow \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$, where $\text{gp}(M)$ is the Grothendieck group of $M$. Let $\mathcal{C}$ be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid $M$ in $\mathcal{C}$ are connected to the com…
Let $R$ be a $G$-graded ring and M be a $G$-graded $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to be the set of all graded primary submodules $Q$ of M such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we define a topology on $\mathcal{PS}_G(M)$ having the Zariski topology on the graded prime spectrum $Spec_G(M)$ as a subspace topology, and investigate several topological properties of this topological space.
Let $\mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $\mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $\mathcal C$-$closed$ if for each congruence $\approx$ on $X$ the quotient semigroup $X/_\approx$ is $\mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\notin\{x,y\}$. We prove that a semigroup…
When $E$ is an $R$-module over a commutative unital ring $R$, the Zariski closure of its support is of the form $\mathrm V(\mathcal O(E))$ where $\mathcal O(E)$ is a unique radical ideal. We give an explicit form of $\mathcal O(E)$ and study its behavior under various operations of algebra. Applications are given, in particular for ring extensions of commutative unital rings whose supports are closed. We provide some applications to crucial and critical ideals of ring extensions.
Let $R$ be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair $(\mathcal{F},\mathcal{I})$ where $\mathcal{F}$ is the class of flat $R$-modules and $\mathcal{I}$ is the class of injective $R$-modules. For a general $R$, the AC-Gorenstein homological algebra of Bravo-Gillespie-Hovey is the one coming from the duality pair $(\mathcal{L},\mathcal{A})$ whe…
Let $R$ be a commutative ring, $\mathcal A$ an $R$-algebra (not necessarily commutative) and $V$ an $R$-subspace or $R$-submodule of $\mathcal A$. By the radical of $V$ we mean the set of all elements $a\in \mathcal A$ such that $a^m\in V$ for all $m\gg 0$. We derive (and show) some necessary conditions satisfied by the elements in the radicals of the kernel of some (partial) differential operators, such as all differential operators of commutative algebras; the differential operators $P(D)$ of (noncommutative) $\mathcal A$ with certain conditions, where $P(\cdot)$ is a polynomial in $n$ commu…
Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in $\mathcal C$. If the endomorphism ring of each module in $\mathcal C$ is semilocal, then $\mathcal V (\mathcal C)$ is a Krull monoid. Although this fact was observed nea…
Let $R$ be a finite commutative ring with $1\ne 0$. The set $\mathcal{F}(R)$ of polynomial functions on $R$ is a finite commutative ring with pointwise operations. Its group of units $\mathcal{F}(R)^\times$ is just the set of all unit-valued polynomial functions, that is the set of polynomial functions which map $R$ into its group of units. We show that $\mathcal{P}_R(R[x]/(x^2))$ the group of polynomial permutations on the ring $R[x]/(x^2)$, consisting of permutations represented by polynomials over $R$, is embedded in a semidirect product of $\mathcal{F}(R)^\times$ by $\mathcal{P}(R)$ the gr…
In this paper, we study the resolving of $\mathcal{SGP}(-)$ and $\mathcal{SGF}(-)$, the classes of all strongly Gorenstein projective and flat modules respectively, over a direct product of commutative rings.
We study Zariski-like topologies on a proper class $X\varsubsetneqq L$ of a complete lattice $\mathcal{L}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the so called classical Zariski topology $(X,τ^{cl})$ and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be $\textit{spectral}$. We say that $\mathcal{L}$ is $X$\emph{-top} iff% \begin{equation*} τ:=\{X\backslash V(a)\mid a\in L\},\text{ where }V(a)=\{x\in L\mid a\leq x\} \end{equation*}% is a topology. We study the interplay between the \textit{algebr…
Let $R$ be an integral domain of characteristic zero, $x=(x_1, x_2, ..., x_n)$ $n$ commutative free variables, and ${\mathcal A}_n:=R[x^{-1}, x]$, i.e., the Laurent polynomial algebra in $x$ over $R$. In this paper we first classify all locally finite or locally nilpotent $R$-derivations and $R$-$\mathcal E$-derivations of ${\mathcal A}_n$, where by an $R$-$\mathcal E$-derivation of ${\mathcal A}_n$ we mean an $R$-linear map of the form $\operatorname{Id}_{{\mathcal A}_n}-ϕ$ for some $R$-algebra endomorphism $ϕ$ of ${\mathcal A}_n$. In particular, we show that ${\mathcal A}_n$ has no nonzero l…
A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, $\scal(R)$ is a spectral space (after Hochster), spectral extension of $\Spec(R)$, and that the assignment $R\mapsto\scal(R)$ induces a contravariant functor. We then relate -- in the case $R$ is an integral domain -- the topology on $\scal(R)$ with the Zariski topology on the set of overrings of $R$. F…
Let $Y$ be an algebraic variety, $\mathcal{F}$ a locally free sheaf of $\mathcal{O}_Y$-modules, and $\mathcal{R}(\mathcal{F})$ the $\mathcal{O}_Y$-algebra $\operatorname{Sym}^\bullet \mathcal{F}$. In this paper we study local properties of sheaves of $\mathcal{O}_{\mathcal{R}(\mathcal{F})}$-ideals $\mathcal{I}$ such that $\mathcal{R}(\mathcal{F}))/\mathcal{I}$ is an algebraic cover of $Y$. Following the work of Miranda for triple covers, for $\mathcal{Q}$ a direct summand of $\mathcal{R}(\mathcal{F})$, we say that a morphism $Φ\colon \mathcal{Q}\rightarrow\mathcal{R}(\mathcal{F})/\langle\mathc…
In this paper, we study the notion of chordality and cycles in hypergraphs from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. However, there is no unified definition for cycle or chordality in hypergraphs in the literature, so we consider several generalizations of these notions and study their algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of hypergraphs. Also we show that if $\mathcal{C}$ is a hypergraph such that $\l…
