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The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different additive subgroups of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations.

In this paper, we introduce the concept of set-valued Prešić–Reich type contractive condition in ultrametric spaces and establish the existence and uniqueness of coincidence and common fixed point of a set-valued and a single-valued mapping besides furnishing illustrative examples to highlight the realized improvements in the context of ultrametric spaces. Our results generalize and extend some known results in the literature.

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

The purpose of this paper is to establish connections between various subclasses of analytic univalent functions by applying certain convolution operator involving Poisson distribution series. To be more precise, we investigate such connections with the classes of analytic univalent functions [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in the open unit disc [Formula: see text].

Let [Formula: see text] be a ring homomorphism and let [Formula: see text] be an ideal of [Formula: see text]. In this paper, we investigate the transfer of the [Formula: see text]-property from a ring [Formula: see text] to his amalgamated algebra [Formula: see text]. Our aim is to give new and original families of [Formula: see text]-rings which are neither [Formula: see text]-rings ([Formula: see text]) nor [Formula: see text]-rings ([Formula: see text]), and examples of [Formula: see text]-rings which are neither [Formula: see text]-rings ([Formula: see text]) nor [Formula: see text]-rings ([Formula: see text]).