In field theory, the expression formula_44 is only shorthand for the formal expression "ab"−1, where "b"−1 is the multiplicative inverse of "b". Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when "b" is zero. Modern texts, that define fields as a special type of ring, include the axiom for fields (or its equivalent) so that the zero ring is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.