Related papers: Powers Vs. Powers

The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare the properties of the stable rank and intrinsic dimension…

Let $R$ be a commutative ring with nonzero identity, and $\delta :\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ the set of all ideals of $R$. In this paper, we introduce the concept of…

Let (R,m) be an n-dimensional regular local ring, essentially of finite type over a field of characteristic zero. In this paper we study the relationship between the singularities of the scheme defined by an m-primary ideal I of R and the…

Inspired by the recent work of Lu and O'Rourke, we study the $a_i$-invariants of (symbolic) powers of some graded ideals. The first scenario is when $I$ and $J$ are two graded ideals in two distinct polynomial rings $R$ and $S$ over a…

In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all…

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We characterize noetherian rings via superstability of the class of left modules with embeddings.…

Let $R$ be a ring, $e$ an idempotent of $R$ and $\delta(R)$ denote the intersection of all essential maximal right ideals of $R$ which is called Zhou radical. In this paper, the Zhou radical of a ring is applied to the $e$-reduced property…

In this study, we present the generalization of the concept of $r$-ideals in commutative rings with nonzero identity. Let $R$ be a commutative ring with $0\neq1$ and $L(R)$ be the lattice of all ideals of $R$. Suppose that…

In this article we study inequalities of ideal norms. We prove that in a subring $R$ of a number field every ideal can be generated by at most $3$ elements if and only if the ideal norm satisfies $N(IJ) \geq N(I)N(J)$ for every pair of…

It has been a well-known fact since Euclid's time that there exist infinitely many rational primes. Two natural questions arise: In which other rings, sufficiently similar to the integers, are there infinitely many irreducible elements? Is…

Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in…

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it…

Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence.…

Motivated by the concept of clean ideals, we introduce the notion of nil clean ideals of a ring. We define an ideal $I$ of a ring $R$ to be nil clean ideal if every element of $I$ can be written as a sum of an idempotent and a nilpotent…

We introduce and investigate a class of ring ideals, termed ring $\mathrm{M}$-ideals, inspired by the Alfsen--Effros theory of $\mathrm{M}$-ideals in Banach spaces. We show that $\mathrm{M}$-ideals extend the classical notion of essential…

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples…

Let $ M (X)$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal. Also, we give several characterizations of maximal ideals of…

In this article, we show that Mori domains, pseudo-valuation domains, and $n$-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain $R$ is a…

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and…

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F(I) of I as reflected in their defining ideals as…