Let $(S,*)$ be a *-semigroup with involution *. Then * satisfies property 

$P$  if $ \forall s,x,y \in S,ss^* = xy^* \Rightarrow s^*x = s^*y.$ We say 

that * is $n$-separable if there is a positive integer $n$ such that 

$ \forall s_1 ,...,s_n  \in S,\exists s_k ,s_i 

\text{such that} s_i s_k^* = s_i s_j^* \Rightarrow s_k  = s_j .

$ Let $(S,*)$ be a *-semigroup with involution * satisfying property $P$. 

Let * be n-proper or $n$-separable. For any positive integer $m$ let $(R,*)

$ be an $mn$-formally complex ring. We prove that







\[

\forall A_1 ,...,A_m  \in R[S]~such~that~\left| {\mathop  \cup \limits_{i = 1}^m 

\sup (A_i )} \right| \le n,\sum\limits_{i = 1}^m {A_i A_i^*  = 0 

\Rightarrow A_i  = 0,i = 1,...,m.} 

\]



In particular if $(T,*)$ is a *-subsemigroup of $S$ such that $\left| T \right| 

\le n$ , then $(R[T],*)$ is $m$-formally complex. Let $(S,*)$ be a *-semigroup. 

We show that if $(Z[S],*)$ is not proper then $(S,*)$ cannot be *-embedded in a 

formally complex ring. We give an example of a finite inverse semigroup $S$ with 

an involution * which is proper, with property $P$, separable and which does not 

coincide with the inverse operator on $S$.