In 2024, Tian proved that a complex matrix [Formula: see text] is Hermitian if and only if [Formula: see text] by means of matrix decomposition. As pointed out by Tian, it would be interesting to extend this result to algebraic settings equipped with a ∗-involution. In this paper, we discuss this result in the setting of an arbitrary ∗-ring, using purely algebraic methods. Let [Formula: see text] be a ∗-ring and [Formula: see text]. Three new properties (P), (Q) and (M) related to Hermitian elements are proposed in order to explore the relationship between [Formula: see text] and [Formula: see text]. It is proved that if [Formula: see text] is ∗-proper ring satisfying property (Q) and [Formula: see text], then [Formula: see text] if and only if [Formula: see text]. Some examples are given to illustrate that property (M) may not necessarily hold for matrices over general fields. Moreover, several new characterizations of Hermitian elements and skew-Hermitian elements in [Formula: see text] are presented.
The classification of idempotent matrices up to similarity is a foundational problem in matrix theory, serving as a bridge between the algebraic properties of rings and the geometric nature of their associated projective modules. A primary objective in this field is to identify the broadest classes of rings, specifically commutative domains, where every idempotent matrix is similar to a standard diagonal idempotent [Formula: see text]. Very recently, Călugăreanu and Lee [Carpathian J. Math. 42 (2026), 613–623] posed the open question of whether this property holds for all commutative GCD domains. While their work established this diagonal reduction for [Formula: see text] idempotents, we give a definitive negative answer for the general case. Let [Formula: see text] be the real affine coordinate ring of the two-sphere. We prove, by Nagata’s criterion, that [Formula: see text] is a unique factorization domain and hence a GCD domain. We explicitly construct an idempotent matrix [Formula: see text] in [Formula: see text]. Its image is the algebraic tangent module of the sphere, which is a finitely generated projective [Formula: see text]-module of rank [Formula: see text]. This module is not free, because freeness would imply the existence of a nowhere-zero continuous tangent vector field on [Formula: see text], contradicting the Hairy Ball Theorem. Therefore, [Formula: see text] is not similar over [Formula: see text] to [Formula: see text], nor to any standard diagonal idempotent. By adjoining zero blocks, this yields counterexamples for all sizes [Formula: see text], thereby resolving the open question in the negative for every [Formula: see text] and showing that the stated [Formula: see text] diagonalization theorem cannot hold under the stated hypotheses. The argument pinpoints the exact module-theoretic obstruction: over domains, diagonal similarity for all idempotents is equivalent to projective-freeness of the coefficient ring, a condition not implied by the GCD property.
We first develope several new homological formulas on semi-trivial ring extensions and in turn construct cotorsion pairs over semi-trivial ring extensions. Finally some applications are given in Morita context rings.
A group [Formula: see text] with finite rank is called an [Formula: see text]-group if every upper central factor of [Formula: see text] is a direct product of an abelian group with the maximal condition on subgroups and a divisible abelian group. Let [Formula: see text] and [Formula: see text] be the inner automorphism group and the group of all central automorphisms of [Formula: see text] fixing the center [Formula: see text] elementwise, respectively. We characterize all [Formula: see text]-groups [Formula: see text] for which [Formula: see text].
In this article, we introduce the semidirect product of the power graphs of two groups. Then we prove that the power graph of the semidirect product of two groups is the semidirect product of their corresponding power graphs. In certain special cases, we provide a necessary and sufficient condition for the non-adjacency of vertices in the product graph.
