Related papers: Graded integral domains which are UMT-domains

We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with…

Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct…

Let R be any ring (with 1), \Gamma a group and R\Gamma the corresponding group ring. Let H be a subgroup of \Gamma of finite index. Let M be an R\Gamma -module, whose restriction to RH is projective. Moore's conjecture: Assume for every…

Building on results of Bazzoni-\v{S}\v{t}ov\'{\i}\v{c}ek, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction…

For a finite group $G$ and an arbitrary commutative ring $R$, Brou\'e has placed a Frobenius exact structure on the category of finitely generated $RG$-modules by taking the exact sequences to be those that split upon restriction to the…

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and…

We give a classification of all irreducible completely pointed $U_q(\mathfrak{sl}_{n+1})$ modules over a characteristic zero field in which $q$ is not a root of unity. This generalizes the classification result of Benkart, Britten and…

Let R be a commutative Noetherian ring. We establish a close relationship between the strong generation of the singularity category of R and the nonvanishing of the annihilator of the singularity category of R. As an application, we prove…

Compared to real-valued signals, complex-valued signals provide a unique and intuitive representation of the phase of real physical systems and processes, which holds fundamental significance and is widely applied across many fields of…

Let $V$ be a rank one valuation domain with quotient field $K$. We characterize the subsets $S$ of $V$ for which the ring of integer-valued polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ is a Pr\"ufer domain. The…

An automorphism on a complex supermanifold $\mathcal M$ is called unipotent if it reduces to the identity on the associated graded supermanifold $gr(\mathcal M)$. These automorphisms are close to be complementary to those responsible for…

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$,…

Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D_{sg}(X) and its idempotent completion, we give necessary…

We will prove that if $G$ and $H$ are modules over a principal ideal domain $R$ such that the endomorphism rings $\mathrm{End}_R(R\oplus G)$ and $\mathrm{End}_R(R\oplus H)$ are isomorphic then $G\cong H$. Conversely, if $R$ is a Dedekind…

Let $R$ be an integral domain. For elements $a,b \in R$, let $[a,b]$ denote their greatest common divisor, if it exists. We say that $R$ has the Z-property if whenever $a,b,c,d$ and $e$ are nonzero nonunits of $R$ such that $abc=de$, then…

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $\Gamma$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded…

We study the topological spectrum of a seminormed ring $R$ which we define as the space of prime ideals $\mathfrak{p}$ such that $\mathfrak{p}$ equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring $R$ we…

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider…

Let $B$ be a totally-definite quaternion algebra over a totally real field $F$, let $\mathfrak{p}$ be a prime ideal of $F$, and let $\Gamma$ be the group of reduced norm-$1$ elements of an Eichler $\mathcal{O}_F[1/\mathfrak{p}]$-order $R$…

Let $\Lambda$ be a $\mathbb{Z}$-graded artin algebra. Two classical results of Gordon and Green state that if $\Lambda$ has only finitely many indecomposable gradable modules, up to isomorphism, then $\Lambda$ has finite representation…