  We consider a class ${\mathcal C}$ of Baer *-rings (also treated in [S. K.
Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195,
Springer-Verlag, Berlin-Heidelberg-New York, 1972.] and [L. Va\v{s}, Dimension
and Torsion Theories for a Class of Baer *-Rings, Journal of Algebra 289 (2005)
no. 2, 614--639]) defined by nine axioms, the last two of which are
particularly strong. We prove that the ninth axiom follows from the first
seven. This gives an affirmative answer to the question of S. K. Berberian if a
Baer *-ring $R$ satisfies the first seven axioms, is the matrix ring $M_n(R)$ a
Baer *-ring.
