Characterizations of Hermitian elements in rings with involution

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.