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### $\left(\delta ,2\right)$-primary ideals of a commutative ring

Czechoslovak Mathematical Journal

Let $R$ be a commutative ring with nonzero identity, let $ℐ\left(ℛ\right)$ be the set of all ideals of $R$ and $\delta :ℐ\left(ℛ\right)\to ℐ\left(ℛ\right)$ an expansion of ideals of $R$ defined by $I↦\delta \left(I\right)$. We introduce the concept of $\left(\delta ,2\right)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $\left(\delta ,2\right)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then ${a}^{2}\in I$ or ${b}^{2}\in \delta \left(I\right)$. Our purpose is to extend the concept of $2$-ideals to $\left(\delta ,2\right)$-primary ideals of commutative rings. Then we investigate the basic properties of $\left(\delta ,2\right)$-primary ideals and also discuss the relations among $\left(\delta ,2\right)$-primary, $\delta$-primary and...

### A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity.

Mathematica Scandinavica

### A class of multiplicative lattices

Czechoslovak Mathematical Journal

We study the multiplicative lattices $L$ which satisfy the condition $a=\left(a:\left(a:b\right)\right)\left(a:b\right)$ for all $a,b\in L$. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $ℤ$ or $ℝ$. A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.

### A class of torsion-free abelian groups characterized by the ranks of their socles

Czechoslovak Mathematical Journal

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket $R$-module is $R$ tensor a bracket group.

### A common generalization of fuzzy primes

Matematički Vesnik

### A criterion for detecting m-regularity.

Inventiones mathematicae

### A full uniform Artin-Rees theorem.

Journal für die reine und angewandte Mathematik

### A generalization of semiflows on monomials

Mathematica Bohemica

Let $K$ be a field, $A=K\left[{X}_{1},\cdots ,{X}_{n}\right]$ and $𝕄$ the set of monomials of $A$. It is well known that the set of monomial ideals of $A$ is in a bijective correspondence with the set of all subsemiflows of the $𝕄$-semiflow $𝕄$. We generalize this to the case of term ideals of $A=R\left[{X}_{1},\cdots ,{X}_{n}\right]$, where $R$ is a commutative Noetherian ring. A term ideal of $A$ is an ideal of $A$ generated by a family of terms $c{X}_{1}^{{\mu }_{1}}\cdots {X}_{n}^{{\mu }_{n}}$, where $c\in R$ and ${\mu }_{1},\cdots ,{\mu }_{n}$ are integers $\ge 0$.

### A graph associated to proper non-small ideals of a commutative ring

Commentationes Mathematicae Universitatis Carolinae

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G\left(R\right)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

### A Note On A Class Of Noetherian Rings

Publications de l'Institut Mathématique

### A note on a class of Noetherian rings.

Publications de l'Institut Mathématique [Elektronische Ressource]

### A note on the algebraic de Morgan's law

Cahiers de Topologie et Géométrie Différentielle Catégoriques

### A note on the intersection multiplicity

Mathematica Slovaca

### A note on unions of ideals and cosets of ideals.

Beiträge zur Algebra und Geometrie

### A realization of $d$-groups

Czechoslovak Mathematical Journal

### A Remark on Rings with Primary Ideals as Maximal Ideals.

Mathematica Scandinavica

### A remark on the spectra of rings with Gabriel dimension

Rendiconti del Seminario Matematico della Università di Padova

### A short proof of Krull's intersection theorem

Colloquium Mathematicae

### A spectral construction of a treed domain that is not going-down

Annales mathématiques Blaise Pascal

### A subspace of $Spec\left(A\right)$ and its connexions with the maximal ring of quotients

Rendiconti del Seminario Matematico della Università di Padova

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