*-Symmetric rings

Let [Formula: see text] be a ∗-ring. Then [Formula: see text] is called a ∗-symmetric ring if for any [Formula: see text] implies [Formula: see text] Obviously, ∗-symmetric ring is certainly symmetric ring, while ∗-ring and symmetric ring does not equal ∗-symmetric ring. Even though reduced rings are symmetric, reduced rings need not be ∗-symmetric ring. In this paper, we further add some conditions for symmetric ring to be ∗-symmetric and reduced. As an application, we discover if [Formula: see text] be a ∗-symmetric ring, then [Formula: see text] Also, we expand the definition of ∗-symmetric ring to different combinations of [Formula: see text]. The properties of ∗-symmetric ring play a role in generalized inverses of elements in rings.