The set "C" contains "A", since any "a" ∈ "A" satisfies the equation "x" − "a" = 0. Moreover, it is possible to prove that "C" is closed under addition and multiplication. Thus, "C" is a subring of "B". The ring "C" is called the [closure of "A" in "B"; or just the integral closure of "A", if "B" is the fraction field of "A"; and the elements of "C" are said to be "integral" over "A". If here formula_12 (the ring of integers) and formula_13 (the field of complex numbers), then "C" is the ring of "algebraic integers".