In this paper, it is shown that a cancellative semigroup is embeddable in an inverse semigroup. It is shown that finite proper *-semigroup is regular and any finite commutative proper *-semigroup is a union of groups. Also it is shown that a finite cyclic proper * semigroup is a group while an infinite one is *-embedded in a proper*-group, and any finite maximal proper*- semigroup has a proper *-extension ring. It is shown that there is a nonregular proper *-ring that cannot be *-embedded in any regular proper *-ring. Also it is shown that an Artinian proper *-ring is a finite direct product of matrix rings over skew fields. It is shown that a commutative proper * and cancellative semigroup is *-embeddable in a regular proper *-semigroup.
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