Related papers: On minimal ring extensions

We study noncommutative rings whose proper subrings all satisfy the same chain condition. We show that if every proper subring of a ring $R$ is right Noetherian, then $R$ is either right Noetherian or the trivial extension of $\mathbb{Z}$…

Let $R$ be a commutative ring with identity $1\neq 0$. In this paper, we continue the study started in [10] concerning when the extended zero-divisor graph of $R$, $\overline{\Gamma}(R)$, is complemented. We also study when…

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…

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the…

In this article we extend the notion of expansivity from topological dynamics to automorphisms of commutative rings with identity. We show that a ring admits a 0-expansive automorphism if and only if it is a finite product of local rings.…

Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is 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…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a uniformly $S$-Artinian ($u$-$S$-Artinian for abbreviation) module if there is $s\in S$ such that any descending chain of…

Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$…

Let $R$ be a commutative noetherian local ring. We define a new invariant for $R$-modules which we call the little dimension. Using it, we extend the improved new intersection theorem.

The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$…

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of $R$ in countable multiplicative subsets. We also show…

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…

Let $R$ be a ring with unity. The upper ideal relation graph $\Gamma_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if…

Let $R$ be a commutative ring and $I$ be an ideal of $R$. The amalgamated duplication of $R$ along $I$ is the subring $R\Join I:=\{(r,r+i)| r\in R, i\in I\}$ of $R\times R$. This paper investigates the extended zero-divisor graph of the…

The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the…

In this work we introduce a new concept, namely, $\tau_{s}$-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show…

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…

Let $R$ be a ring with involution containing a nontrivial symmetric idempotent element $e$. Let $\delta: R\rightarrow R$ be a mapping such that $\delta(ab)=\delta(b)a^{\ast}+b^{\ast}\delta(a)$ for all $a,b\in R$, we call $\delta$ a…

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology.…