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Ideal Theory in $f$-Algebras

C. B. Huijsmans and B. de Pagter
Transactions of the American Mathematical Society
Vol. 269, No. 1 (Jan., 1982), pp. 225-245
DOI: 10.2307/1998601
Stable URL: http://www.jstor.org/stable/1998601
Page Count: 21
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Ideal Theory in $f$-Algebras
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Abstract

The paper deals mainly with the theory of algebra ideals and order ideals in $f$-algebras. Necessary and sufficient conditions are established for an algebra ideal to be prime, semiprime or idempotent. In a uniformly complete $f$-algebra with unit element every algebra ideal is an order ideal $\operatorname{iff} the $f$-algebra is normal. This result is based on the fact that the range of every orthomorphism in a uniformly complete normal Riesz space is an order ideal.

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