Related papers: Jeff Smith's Theory of Ideals
Pursuing ideas of Jeff Smith, we develop a homotopy theory of ideals of monoids in a symmetric monoidal model category. This includes Smith ideals of structured ring spectra and of differential graded algebras. Such Smith ideals are NOT…
For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…
We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial…
Symmetric spectra were introduced by Jeff Smith as a symmetric monoidal category of spectra. In this paper, a detection functor is defined which detects stable equivalences of symmetric spectra. This detection functor is useful because the…
We define the spectrum of a tensor triangulated category $K$ as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects…
We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…
An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$…
In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-ideals and examine…
This article is concerned with the number of generators of perfect ideals J in regular local rings (R,m). If J is sufficiently large modulo $m^n$, a bound is established depending only on n and the projective dimension of J. More ambitious…
The regular graph of ideals of the commutative ring $R$, denoted by ${\Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$…
This note remarks that the correspondence between non-unital algebras and augmented unital algebras can be derived from Hovey's Smith ideal theory. Applying Smith ideal theory of stable symmetric monoidal model category, we formulate…
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 begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how…
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…
Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and…
The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb Q[x_1, \dots, x_n]$ to a corresponding ideal in $\mathbb F_p[x_1,\dots, x_n]$ where $p$ is a prime number; in other words, the…
Let $\mathcal{I}(R)$ be the set of all ideals of a ring $R$, $\delta$ be an expansion function of $\mathcal{I}(R)$. In this paper, the $\delta$-$J$-ideal of a commutative ring is defined, that is, if $a, b\in R$ and $ab\in I\in…
Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly $J$-ideals as a new generalization of $J$-ideals. We call a proper ideal $I$ of a ring $R$ a weakly $J$-ideal if whenever $a,b\in R$…
The concept of $\Gamma$-semigroups was introduced by M. K Sen in 1981. This study aims to investigate several intriguing properties of $\Gamma$-semigroups and to provide the concepts of simple $\Gamma$-semigroups, 0-simple…
In this paper homotopical methods for the description of subgroups determined by ideals in group rings are introduced. It is shown that in certain cases the subgroups determined by symmetric product of ideals in group rings can be described…