Vol. 27, No. 1, 1968

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On rings with proper involution

Willard Ellis Baxter

Vol. 27 (1968), No. 1, 1–12

A topological ring is said to have property (Y ) if, and only if, 2A = A; A has a proper continuous involution (with symmetric elements S) such that whenever the net {2xα} tends to zero, so also does {xα}; A3 is dense in A; and the left annihilator of a closed Jordan ideal, U, of S is zero if, and only if, U is S.

One shows that for such rings and for annihilator rings with the first two properties above that every closed Jordan ideal of S is the intersection of S with a closed two-sided ideal. Also shown is the fact that S S is dense in S.

A study is made of relations between the socle and Jordan ideals of S for topological rings. Finally, a new proof of Herstein’s result for S in simple associative rings is given.

Mathematical Subject Classification
Primary: 16.98
Secondary: 46.00
Received: 11 August 1967
Published: 1 October 1968
Willard Ellis Baxter