Let R be a unitary ring without zero divisors. We prove that if R contains an element with finite centralizer, then R must be commutative. Furthermore, employing Zorn’s Lemma (which is equivalent to the Axiom of Choice), we demonstrate that in the noncommutative case, every element of R is contained in an infinite commutative subring.

For a continuous self-map of a compact metric space, we provide a sufficient condition for the orbit of a point to converge to a periodic orbit or an odometer. We show that if a continuous self-map of a compact metric space has the shadowing property, then the set of points whose orbits converge to periodic orbits or odometers is dense.

We endow the set of all invariant measures of a topological dynamical system with a metric $\barρ$, which induces a topology stronger than the the weak$^*$-topology. Then, we study the closedness of ergodic measures within a characteristic class under this metric. Specifically, we show that if a sequence of generic points associated with ergodic measures from a fixed characteristic class converges in the Besicovitch pseudometric, then the limit point is generic for an ergodic measure in the same class. This implies that the set of ergodic measures belonging to a fixed characteristic class is closed in $\barρ$ (by a result of Babel, Can, Kwietniak, and Oprocha in [1]).

We present a simple geometric approach to studying the $L^p$ boundedness properties of Stein’s spherical maximal operator, which does not rely on the Fourier transform. Using this, we recover a weak form of Stein’s spherical maximal theorem.

We initiate the study of cortex algebras (commutative Banach algebras with extendable multiplicative-linear functionals) and the spectral permanence property. In addition, we analyze some particular examples in this context and present open problems.

For two neighboring stiff inclusions, the stress, which is the gradient of a solution to the Lamé system of linear elasticity, may exhibit singular behavior as the distance between these two inclusions becomes arbitrarily small. The main contribution of t

Let $E$ be an atomic reflexive Banach lattice and $X$ be any Banach lattice with Pełczyński’s property (V). We show that (i) the positive injective tensor product $E\mathbin {\check {\otimes }_{|\varepsilon |}}X$ has Pełczyński’s property (V); (ii) the po

The results that are stated in [Colloq. Math. 162 (2020), 279–301] hold true but due to an oversight by the author in the publication process, the definition of a system being left/right non-degenerate was stated incorrectly. The purpose of this note is t

In the current paper we attempt to transfer the notion of the projectional entropy, originally defined for multidimensional subshifts, to the case of actions of amenable groups. The main theorem states that if a system is strongly irreducible the equality of the entropy and the projectional entropy implies that the system has a product-like structure.