Related papers: On purely-prime ideals with applications

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ of characteristic $p$, with Deligne--Lusztig dual $G^\ast$. We show that, over $\overline{\mathbb{Z}}[1/pM]$ where $M$ is the product of all bad primes for…

In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a…

The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched…

Let $R$ be a commutative $G$-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal $I$ of $R$ is said to be graded radically principal if $Grad(I)=Grad(\langle…

This article mentions that Smith ideal theory generalizes the adic completion theory of commutative rings to monoid objects of locally presentable symmetric monoidal abelian categories. As an application, we provide an almost mathematics…

A complete understanding of the structure of all prime ideals of an infinite direct product of commutative rings (e.g. in terms of more specific objects) has remained a challenging problem for decades. In this article, new advances have…

Consider the polynomial ring $R_n = k[x_1,...,x_n]$, where $k$ is a field. Let $m = (x_1,...,x_n)$ and $I$ be an $m$-primary monomial ideal in $R$. We consider the problem of determining whether such ideals are in the Gorenstein liasion…

We determine criteria for the prime spectrum of an ambiskew polynomial algebra $R$ over an algebraically closed field $K$ to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra…

If $R$ is a topological ring then $R^{\ast}$, the group of units of $R$, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By an \emph{absolute topological ring} we mean a…

In this paper, leveraging the recent achievements of researchers, we have revisited the family of ideals of product of commutative rings. We demonstrate that if $ \{ R_\alpha \}_{\alpha \in A} $ is an infinite family of rings, then $ \left|…

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…

We study prime ideals in skew power series rings $T:=R[[y;\tau,\delta]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $\tau$ of $R$, and $\tau$-derivations $\delta$ of $R$. These rings were…

We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points…

Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most…

In this article, we introduce and study the concept of $\phi$-2-absorbing quasi primary ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $L(R)$ be the lattice of all ideals of $R$. Suppose that…

An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper…

Some descriptions of linked ideals in a commutative Notherian ring $R$ are provided in terms of the Associated prime ideals of $R$. Then, among other things, we make some characterization of Cohen-Macaulay, Gorenstein and regular local…

In this paper, we define a new concept of Noetherian commutative rings which stands between Gorenstein and Cohen-Macaulay properties. We show that this new property keep hold under common operations of commutative rings such as…

Let R be a commutative ring. If P is a maximal ideal of R whose a power is finitely generated then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension…