On Unitification of $\ast$-rings

S. K. Berberian raised the open problem ``Can every weakly Rickart $\ast$-ring be embedded in a Rickart $\ast$-ring? with preservation of right projections?" Berberian has given a partial solution to this problem. Khairnar and Waphare raised a similar problem for p.q.-Baer $\ast$-rings and gave a partial solution. In this paper, we give more general partial solutions to both the problems.