Annihilator conditions on modules over commutative rings

Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-module. Let [Formula: see text] and [Formula: see text]. [Formula: see text] satisfies Property [Formula: see text] (respectively, Property [Formula: see text]) if for each finitely generated ideal [Formula: see text] (respectively, finitely generated submodule [Formula: see text]) ann[Formula: see text] (respectively, ann[Formula: see text]). The ring [Formula: see text] satisfies Property [Formula: see text] if [Formula: see text] does. We study rings and modules satisfying Property [Formula: see text] or Property [Formula: see text]. A number of examples are given, many using the method of idealization.

Linear dynamics and recurrence properties defined via essential idempotents of

Consider $\mathscr{F}$, a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property ${\mathcal{P}}_{\mathscr{F}}$ if, for any non-empty open set $U$ in $X$, there exists $x\in X$ such that $\{n\geq 0:T^{n}x\in U\}\in \mathscr{F}$. Let $\overline{{\mathcal{B}}{\mathcal{D}}}$ be the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of a sequence of operators satisfying property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$, for which we have used a deep result of Bergelson and McCutcheon in the vein of Szemerédi’s theorem. It turns out that operators having property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\unicode[STIX]{x1D6FD}\mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.

On O’Malley preponderantly continuous functions

In the paper some algebraic properties of functions which are preponderantly continuous in O'Malley sense and functions satisfying property

A General Coincidence and Common Fixed Point Theorem for two Hybrid Pairs of Mappings

In this paper, a general coincidence and common fixed points theorem for two hybrid pairs of mappings satisfying property (E.A) is proved, generalizing the main results from [3] and [9].

SOLUTION TO $\overline{\partial}$-EQUATIONS WITH EXACT SUPPORT ON PSEUDO-CONVEX MANIFOLDS

We obtain a solution to the [Formula: see text]-equation with exact support in a domain Ω with C1-smooth boundary satisfying property B in a complex manifold. This is done for complex-valued forms of type (r,s), s ≥ 1 and for forms of type (r,s), q ≤ s ≤ n - q, with values in a holomorphic vector bundle when the domain Ω is strongly q-convex.