To be treated as an addition to all current field axioms
"For every A in S there exists a z1 and a z2 constituting A. Such that any A in operation of a binary expression of multiplication or division is only representing z1 or z2. Such that z1 and z2 for all A's other than zero equal A. Such that z1 for zero equals zero. Such that z2 for zero equals 1."
Note....only a z1 and a z2 may exist in any binary expression of multiplication or division.
Note.... either symbol(number) in a binary expression of multiplication can be labeled z1 or z2. In a binary expression of division z1 must come first, while z2 must come second.
This is an attempt to allow for....
division by zero.
an inverse operation for multiplication by zero, other than zero as the sum.