Ideal theory in 𝑓-algebras

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

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

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Ideal theory in -algebras

Authors: C. B. Huijsmans and B. de Pagter
Journal: Trans. Amer. Math. Soc. 269 (1982), 225-245
MSC: Primary 06F25; Secondary 46A40, 46J20, 54C40
MathSciNet review: 637036
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Abstract: The paper deals mainly with the theory of algebra ideals and order ideals in -algebras. Necessary and sufficient conditions are established for an algebra ideal to be prime, semiprime or idempotent. In a uniformly complete -algebra with unit element every algebra ideal is an order ideal iff the -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.

[1] Ichirô Amemiya, A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser. I. 12 (1953), 111–156. MR 0056853 (15,137d)
[2] Eleanor R. Aron and Anthony W. Hager, Convex vector lattices and 𝑙-algebras, Topology Appl. 12 (1981), no. 1, 1–10. MR 600458 (82c:54010), http://dx.doi.org/10.1016/0166-8641(81)90024-9
[3] S. J. Bernau, On semi-normal lattice rings, Proc. Cambridge Philos. Soc. 61 (1965), 613–616. MR 0183656 (32 #1136)
[4] Alain Bigard, Les orthomorphismes d’un espace réticulé archimédien, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34 (1972), 236–246 (French). MR 0308000 (46 #7115)
[5] Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin, 1977 (French). MR 0552653 (58 #27688)
[6] Garrett Birkhoff and R. S. Pierce, Lattice-ordered rings, An. Acad. Brasil. Ci. 28 (1956), 41–69. MR 0080099 (18,191d)
[7] László Fuchs, Teilweise geordnete algebraische Strukturen, Studia Mathematica–Mathematische Lehrbücher, Band XIX. Übersetzt aus dem Englischen von Éva Vas, Vandenhoeck & Ruprecht, Göttingen, 1966 (German). MR 0204547 (34 #4386)
[8] L. Gillman, Rings with Hausdorff structure space, Fund. Math. 45 (1957), 1–16. MR 0092773 (19,1156a)
[9] Leonard Gillman and Melvin Henriksen, Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. 82 (1956), 366–391. MR 0078980 (18,9d), http://dx.doi.org/10.1090/S0002-9947-1956-0078980-4
[10] Leonard Gillman and Meyer Jerison, Rings of continuous functions, Springer-Verlag, New York, 1976. Reprint of the 1960 edition; Graduate Texts in Mathematics, No. 43. MR 0407579 (53 #11352)
[11] Leonard Gillman and Carl W. Kohls, Convex and pseudoprime ideals in rings of continuous functions, Math. Z. 72 (1959/1960), 399–409. MR 0114115 (22 #4942)
[12] Melvin Henriksen, Semiprime ideals of 𝑓-rings, Symposia Mathematica, Vol. XXI (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), Academic Press, London, 1977, pp. 401–409. MR 0480256 (58 #435)
[13] M. Henriksen, J. R. Isbell, and D. G. Johnson, Residue class fields of lattice-ordered algebras, Fund. Math. 50 (1961/1962), 107–117. MR 0133350 (24 #A3184)
[14] M. Henriksen and D. G. Johnson, On the structure of a class of archimedean lattice-ordered algebras., Fund. Math. 50 (1961/1962), 73–94. MR 0133698 (24 #A3524)
[15] C. B. Huijsmans, Some analogies between commutative rings, Riesz spaces and distributive lattices with smallest element, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 132–147. MR 0354635 (50 #7113)
[16] C. B. Huijsmans and B. de Pagter, On 𝑧-ideals and 𝑑-ideals in Riesz spaces. II, Nederl. Akad. Wetensch. Indag. Math. 42 (1980), no. 4, 391–408. MR 597997 (83c:46004a)
[17] D. G. Johnson, A structure theory for a class of lattice-ordered rings, Acta Math. 104 (1960), 163–215. MR 0125141 (23 #A2447)
[18] E. de Jonge and A. C. M. van Rooij, Introduction to Riesz spaces, Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 78. MR 0473777 (57 #13439)
[19] W. A. J. Luxemburg, Some aspects of the theory of Riesz spaces, University of Arkansas Lecture Notes in Mathematics, vol. 4, University of Arkansas, Fayetteville, Ark., 1979. MR 568706 (83f:46010)
[20] W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. I, North-Holland, Amsterdam and London, 1971.
[21] Hidegorô Nakano, Modern Spectral Theory, Maruzen Co. Ltd., Tokyo, 1950. MR 0038564 (12,419f)
[22] David Rudd, On two sum theorems for ideals of 𝐶(𝑋), Michigan Math. J. 17 (1970), 139–141. MR 0259616 (41 #4252)
[23] G. L. Seever, Measures on 𝐹-spaces, Trans. Amer. Math. Soc. 133 (1968), 267–280. MR 0226386 (37 #1976), http://dx.doi.org/10.1090/S0002-9947-1968-0226386-5
[24] H. Subramanian, 𝑙-prime ideals in 𝑓-rings, Bull. Soc. Math. France 95 (1967), 193–203. MR 0223284 (36 #6332)
[25] R. Douglas Williams, Intersections of primary ideals in rings of continuous functions, Canad. J. Math. 24 (1972), 502–519. MR 0295066 (45 #4134)

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