Related papers: On $p-$Ring

In this paper, we introduce the notion of quasi-$F$-splitting for rings in mixed characteristic. By comparing quasi-$F$-splitting with perfectoid purity, we obtain a new inversion of adjunction-type result. Furthermore, we study the…

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$. In this paper, we extend Hartshorne's characterization of cofinite complexes to more general classes of rings. We also determine conditions under which…

In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that…

Let $f: A\rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we investigate the transfer of self-injective property to the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by…

Let $R$ be a commutative ring with identity and $\Bbb A (R)$ be the set of ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$, denoted by $A_{I} (R) $, is a simple graph with the vertex set $\Bbb A(R)^{\ast} := \Bbb…

The authors introduced the notion of $p_g$-ideals for two-dimensional excellent normal local domain over an algebraicaly closed field in terms of resolution of singularities. In this note, we give several ring-theoretic characterization of…

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…

We define a Jordan homomorphism $\varphi$ from a ring $R$ to a ring $R'$ to be splittable if the ideal (of the subring generated by the image of $\varphi$) generated by all $\varphi(xy)-\varphi(x)\varphi(y)$, $x,y\in R$, has trivial…

In a ring $A$ an ideal $I$ is called (principally) nilary if for any two (principal) ideals $V, W$ in $A$ with $VW\subseteq I,$ then either $V^n\subseteq I$ or $W^m\subseteq I,$ for some positive integers $m$ and $n$ depending on $V$ and…

Let $P$ be a preordered set, $R$ a ring and $FI(P,R)$ the finitary incidence ring of $P$ over $R$. We find a criterion for all Jordan derivations of $FI(P,R)$ to be derivations and generalize Theorem 3.3 from arXiv:1411.6123. In particular,…

Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of \textit{$D$-valued $B$-rational functions on $E$}, denoted by $Int^R_B(E,D)$, which naturally extends the…

For a number field $K$, we extend the notion of the ring class field of an order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in $K$. We give both ideal-theoretic and idele-theoretic description of…

Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $\Phi$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $\Phi(N)=I(R,N)RG$ where $I(R,N)$ is…

For $a\in R$, let $P_a$ denote the intersection of all minimal prime ideals of $R$ containing $a$. An ideal $I$ of a ring $R$ is called a $z^{\circ}$-ideal if $P_a\subseteq I$ for all $a\in I$. In this paper, we first investigate the class…

A cover by left ideals of an associative (not necessarily commutative or unital) ring $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, then $\eta_\ell(R)$ is the cardinality of a…

Necessary and sufficient conditions for an Ore extension $S=R[x;\si,\de]$ to be a {\rm PI} ring are given in the case $\si$ is an injective endomorphism of a semiprime ring $R$ satisfying the {\rm ACC} on annihilators. Also, for an…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains…

Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and…

It is shown that a commutative B\'ezout ring $R$ with compact minimal prime spectrum is an elementary divisor ring if and only if so is $R/L$ for each minimal prime ideal $L$. This result is obtained by using the quotient space…