Class of Baer ∗-rings defined by a relaxed set of axioms

We consider a class C of Baer ∗-rings (also treated in [S.K. Berberian, Baer ∗-Rings, Grundlehren Math. Wiss., vol. 195, Springer, Berlin, 1972] and [L. Vaš, Dimension and torsion theories for a class of Baer ∗-rings, J. Algebra 289 (2005) 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 Mn(R) a Baer ∗-ring.