Related papers: Reduced points of $\mathbb{E}_{\infty}$-rings in p…

In this paper we provide necessary and sufficient conditions for $ R=A\propto E $ to be a valuation ring where $E$ is a non-torsion or finitely generated $A-$module. Also, we investigate the $ (n,d) $ property of the valuation ring.

We consider the class of all commutative reduced rings for which there exists a finite subset T of A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for…

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the…

We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is,…

In this article, we consider the structure of graded rings, not necessarily commutative nor with unity, and study the graded weakly prime ideals. We investigate the graded rings in which all graded ideals are graded weakly prime. Several…

For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I…

Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an aleph_1-free R-module G of rank aleph_1 with endomorphism algebra End_RG=A . Clearly…

A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits…

Given a reduced, local ring $R$ and an ideal $\mathfrak{a}$ of positive height, we give a decomposition of the test module, $\tau(\omega_T, t^{-\lambda})$, of the extended Rees algebra, $T =R[\mathfrak{a} t, t^{-1}]$. In particular, the…

Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $\Delta$ is an infinite set of positive integers, such that…

Let $A$ be an $\mathbf{E}_{\infty}$-ring spectrum over the rational numbers. If $A$ satisfies a noetherian condition on its homotopy groups $\pi_*(A)$, we construct a collection of $\mathbf{E}_{\infty}$-$A$-algebras that realize on homotopy…

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a…

When the base ring is not a field, power reductivity of a group scheme is a basic notion, intimately tied with finite generation of subrings of invariants. Geometric reductivity is weaker and less pertinent in this context. We give a survey…

It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p)…

Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It…

This paper introduces and studies nil-reversible rings wherein we call a ring R nil-reversible if the left and right annihilators of every nilpotent element of R are equal. Reversible rings (and hence reduced rings) form a proper subclass…

We raise the following general question regarding a ring graded by a group: "If $P$ is a ring-theoretic property, how does one define the graded version $P_{\operatorname{gr}}$ of the property $P$ in a meaningful way?". Some properties of…

Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p > 0$. This paper continues a long-standing effort to classify the connected reductive subgroups of $G$. Having previously…

We show that if a closed manifold M admits an F-structure (possibly of rank 0) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial circle action. As a corollary we obtain that the simplicial volume…