On S-torsion exact sequences and Si-projective modules (i = 1,2)

Let [Formula: see text] be a commutative ring and [Formula: see text] a given multiplicative closed subset of [Formula: see text]. In this paper, we introduce the new concept of [Formula: see text]-torsion exact sequences (respectively, [Formula: see text]-torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules.

Best proximity point theorems for $ \alpha^{+} F, (\theta-\phi )$-proximal contraction

In this paper, inspired by the idea of Suzuki type $ \alpha^{+} F$-proximal contraction in metric spaces, we prove a new existence of best proximity point for Suzuki type $ \alpha^{+} F$-proximal contraction and $ \alpha^{+} (\theta-\phi )$-proximal contraction defined on a closed subset of a complete metric space. Our theorems extend, generalize, and improve many existing results.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

SOME CHARACTERIZATIONS OF DICHOTOMY FOR IMPULSIVE DYNAMIC SYSTEMS

We study the problem of dichotomy and boundedness for impulsive dynamic equations on arbitrary closed subset of real numbers. The spectral decomposition theorem gives all our main results. The obtained results are fundamentally new, even for the classical case.

When is (D, K) an S-accr pair?

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLet S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.Originality/valueMotivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

On Weakly S-Primary Ideals of Commutative Rings

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

Pairs of domains where all intermediate domains satisfy S-ACCP

The rings considered in this paper are commutative with identity. If [Formula: see text] is a subring of a ring [Formula: see text], then we assume that [Formula: see text] contains the identity element of [Formula: see text]. Let [Formula: see text] be a multiplicatively closed subset (m.c. subset) of a ring [Formula: see text]. An increasing sequence of ideals [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-stationary if there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This paper is motivated by the research work [A. Hamed and H. Kim, On integral domains in which every ascending chain on principal ideals is [Formula: see text]-stationary, Bull. Korean Math. Soc. 57(5) (2020) 1215–1229]. Let [Formula: see text] be a m.c. subset of an integral domain [Formula: see text]. We say that [Formula: see text] satisfies [Formula: see text]-ACCP if every increasing sequence of principal ideals of [Formula: see text] is [Formula: see text]-stationary. Let [Formula: see text] be a subring of an integral domain [Formula: see text] and let [Formula: see text] be a m.c. subset of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-ACCP pair if [Formula: see text] satisfies [Formula: see text]-ACCP for every subring [Formula: see text] of [Formula: see text] with [Formula: see text]. The aim of this paper is to provide some pairs of domains [Formula: see text] such that [Formula: see text] is an [Formula: see text]-ACCP pair, where [Formula: see text] is a m.c. subset of [Formula: see text].

The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

Frum-Ketkov type multivalued operators

Let (M,d) be a metric space, X\subset M be a nonempty closed subset and K\subset M be a nonempty compact subset. By definition, an upper semi-continuous multivalued operator F:X\to P(X) is said to be a strong Frum-Ketkov type operator if there exists \alpha\in ]0,1[ such that e_d(F(x),K)\le \alpha D_d(x,K), for every x\in X, where e_d is the excess functional generated by d and D_d is the distance from a point to a set. In this paper, we will study the fixed points of strong Frum-Ketkov type multivalued operators.