Narrow Results

Let $R$ be a commutative ring with identity and $S$ be a multiplicative closed subset of $R$. In this paper, we introduce and study the concept of strongly $S$-1-absorbing primary ideals as subclass of primary ideals. We define a proper ideal $I$ disjoint with $S$ to be a strongly $S$-1-absorbing primary ideal if there exists an $s\in S$ such that for all nonunit elements $a,b,c\in R$ and $abc\in I$, we have $sab\in I$ or $sc\in \sqrt{0}$. We characterize local rings and UN-rings with the help of strongly $S$-1-absorbing primary ideals. Furthermore, we investigate the transfer of the above properties to trivial ring extensions, amalgamations and bi-amalgamations of rings along an ideal.

This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.

Recently, Xiang and Ouyang defined a (commutative unital) ring $R$ to be Nil${}_{*}$-coherent if each finitely generated ideal of $R$ that is contained in Nil$(R)$ is a finitely presented $R$-module. We define and study Nil${}_{*}$-coherent modules and special Nil${}_{*}$-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil${}_{*}$-coherent ring and any special Nil${}_{*}$-coherent ring is a Nil${}_{*}$-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil${}_{*}$-coherent ring (regardless of whether it is coherent). Several examples of Nil${}_{*}$-coherent rings that are not special Nil${}_{*}$-coherent rings are obtained as byproducts of our study of the transfer of the Nil${}_{*}$-coherent and the special Nil${}_{*}$-coherent properties to trivial ring extensions and amalgamated algebras.

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.

Let $R$ be a commutative ring with $1\ne 0$. Let $n\ge 1$ be a positive integer. A proper ideal $I$ of $R$ is called an n-absorbing ideal (respectively, a strongly n-absorbing ideal) of $R$ as in [D. F. Anderson and A. Badawi, On $n$-absorbing ideals of commutative rings, Comm. Algebra39 (2011) 1646–1672] if $a_1,a_2,\dots ,a_{n+1}\in R$ and $a_1.a_2\dots a_{n+1}\in I$, then there are $n$ of the $a_i$’s whose product is in $I$ (respectively, if whenever $I_1\dots I_{n+1}\subseteq I$ for ideals $I_1,\dots ,I_{n+1}$ of $R$, then the product of some $n$ of the $I_j^{\prime }$s is contained in $I$). The concept of $n$-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ is a 1-absorbing ideal of $R$). Let $f:A\to B$ be a ring homomorphism and let $J$ be an ideal of $B.$ This paper investigates the $n$-absorbing and strongly $n$-absorbing ideals in the amalgamation of $A$ with $B$ along $J$ with respect $f$ denoted by $A{\bowtie }^{f}J.$ The obtained results generate new original classes of $n$-absorbing and strongly $n$-absorbing ideals.

Let $f:A\to B$ be a ring homomorphism, $J$ be an ideal of $B$ and $A{\bowtie }^{f}J$ the amalgamation of $A$ with $B$ along $J$ with respect to $f$. In this paper, we provide necessary and sufficient conditions for $A{\bowtie }^{f}J$ to be a Steinitz ring, semi-Steinitz ring, and weakly semi-Steinitz. Then we construct new original examples of weakly semi-Steinitz rings which are not semi-Steinitz rings and semi-Steinitz rings which are not Steinitz rings.

In this paper, we extend the notion of 2-nil ideal introduced by Yetkin Celikel in [E. Yetkin Celikel, 2-nil ideals of commutative rings, Bulletin of the Belgian Mathematical Society Simon Stevin 28 (3) (2021).] to 2-nil submodule which is a subclass of 2-absorbing primary submodules. Let $M$ be an $R$-module and $N$ be a proper submodule of $M$. We say that $N$ is a 2-nil submodule of $M$ if whenever $a,b\in R$, $m\in M$ and $abm\in N$, then $ab\in \mathrm{\text{Nil}}(M)$ or $am\in N$ or $bm\in N.$ We study the properties of this concept and establish several characterizations. We also investigate the 2-nil ideals of amalgamation. The obtained results yield new original families of examples of 2-nil ideals.

Let $A={\oplus }_{\alpha \in \mathrm{\Gamma }}{A}_{\alpha }$ and $B={\oplus }_{\alpha \in \mathrm{\Gamma }}{B}_{\alpha }$ be two commutative rings graded by an arbitrary commutative monoid $\mathrm{\Gamma }$, let $J$ be a homogeneous ideal of $B$ and $f:A\to B$ a graded ring homomorphism. In this paper, we introduce a new graded ring construction called “the graded amalgamation of $A$ with $B$ along $J$ with respect to $f$” (denoted by $A{\bowtie }^{f}J$). Then we investigate, through the properties of the homogeneous components that build the body of this “graded” study, the transfer of the property of graded-coherence to the graded amalgamation $A{\bowtie }^{f}J$.

Let $f:A\to B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we investigate the transfer of the $(n,d)$-property from a ring $A$ to his amalgamated algebra $A{\bowtie }^{f}J$. Our aim is to give new and original families of $(2,d)$-rings which are neither $(1,d)$-rings ($d=0,1,2$) nor $(2,d-1)$-rings ($d=1,2$), and examples of $(3,d)$-rings which are neither $(2,d)$-rings ($d\ge 0$) nor $(3,d-1)$-rings ($d\ge 1$).

Let $R$ and $S$ be two commutative rings with unity, let $J$ be an ideal of $S$ and $\phi :R\to S$ be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra $R{\bowtie }^{\phi }J$ to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over $R{\bowtie }^{\phi }J$ has a complete comaximal factorization.

In this paper, we introduce a generalization of the well-known notion of weak coherent rings, which we call weak $P$-coherent rings. We investigate the stability of this property under direct products and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions and amalgamated algebras along an ideal. We conclude with various examples of weak $P$-coherent rings.