Vol. 33, No. 1, 1970

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On normed rings with monotone multiplication

Silvio Aurora

Vol. 33 (1970), No. 1, 15–20

It is shown that if a normed division ring has a norm which is “multiplication monotone” in the sense that N(x) < N(x) and N(y) < N(y) imply N(xy) N(xy), and if the norm is “commutative” in the sense that N(xy) = N(yx) for all x and y, then the topology of that ring is given by an absolute value. A consequence of this result is that if the norm of a connected normed ring with unity is multiplication monotone and commutative then the ring is embeddable in the system of quaternions.

Mathematical Subject Classification
Primary: 46.50
Received: 9 April 1969
Published: 1 April 1970
Silvio Aurora