*-Symmetric rings

Let $R$ be a ∗-ring. Then $R$ is called a ∗-symmetric ring if for any $a,b,c\in R,abc=0$ implies $ac{b}^{\ast }=0.$ 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 $R$ be a ∗-symmetric ring, then $R^{\#}=R^{EP}=R^{+}=R^{\mathrm{\text{reg}}}.$ Also, we expand the definition of ∗-symmetric ring to different combinations of $a,b,c$. The properties of ∗-symmetric ring play a role in generalized inverses of elements in rings.