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It is shown that if a normed division ring has a norm which is "multiplicati on monotone" in the sense that N(x) 0 for x φ 0, (ii) N(-x) = N(x) for all x, (iii) N(x + y) ^ N(x) + N(y) for all x and y, (iv) N(xy) < N(x)N(y) for all x and y.) Ostrowski's results may be regarded as the treatment of the special case of this problem in which the norm N satisfies the additional condition N(xy) = N(x)N(y) for all x and y. This extra requirement is replaced here by the weaker condition that N be multiplication monotone in the sense that whenever N(x) < N(x') and N(y) < N(y f) then N(xy) ^ N(x'y'). Specifically, it is shown in the corollary of Theorem 3 that if a commutative connected normed ring with unity has a multiplication monotone norm then that ring is (algebraicall y and topologically isomorphic to) a subring of the field of complex numbers. (The version of this statement which appears below actually includes the noncommutative case as well.) The basic device employed in obtaining this result is Theorem 2, which asserts that if a normed division ring has a multiplication monotone norm N such that

On normed rings with monotone multiplication