* weak e-symmetric rings

Let [Formula: see text] be a *-ring and [Formula: see text]. Then [Formula: see text] is called a * weak [Formula: see text]-symmetric ring if for any [Formula: see text] implies [Formula: see text] It is shown that * weak [Formula: see text]-symmetric rings are weak [Formula: see text]-symmetric. However, the converse is not true. We show that for a *-ring [Formula: see text] with [Formula: see text], if [Formula: see text] is a weak [Formula: see text]-symmetric ring and a * [Formula: see text]-cancellable ring, then [Formula: see text] is * weak [Formula: see text]-symmetric. Also, many properties of * weak [Formula: see text]-symmetric rings are discussed.