Ideal theory in 𝑓-algebras

Huijsmans, C. B.; de Pagter, B.

doi:10.1090/S0002-9947-1982-0637036-5

Ideal theory in $f$-algebras
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by C. B. Huijsmans and B. de Pagter

Trans. Amer. Math. Soc. 269 (1982), 225-245

DOI: https://doi.org/10.1090/S0002-9947-1982-0637036-5

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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 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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