Some equivalents of AC in algebra

It is shown that the axiom of choice AC is equivalent to the statements: (1)For every Boolean ring (B, +, ·)and every subset H B which is closed under + there exists a –maximal ideal QB such that H Q={0} and, (2)For every Boolean ring (B,+,·),for every A B, the infinite system:X _i + y_i=b_i,iI, b _i B has a solution in A iff each of its equations has a solution in A.Presented by G. Grätzer