Integer-Valued Continuous Functions II

Subramanian, H.

Canadian mathematical bulletin, Tome 14 (1971) no. 2, pp. 235-237

Résumé

We follow [6] and [7] for all terminologies. The purpose of this note is to proveTheorem 1. Let X and Y be any two integer-compact spaces. The following are equivalent: (1) X is homeomorphic to Y. (2) C(X, Z) and C(Y, Z) are isomorphic as rings. (3) C(X, Z) and C(F, Z) are isomorphic as lattices. (4) C(X, Z) and C(Y, Z) are isomorphic as p.o. groups. (5) C(X, Z) and C(Y, Z) are isomorphic as multiplicative semigroups.

Détail

DOI : 10.4153/CMB-1971-040-2

Bibliographie

[1] 1. Ailing, N. L., Rings of continuous integer-valued functions and nonstandard arithmetic, Trans. Amer. Math. Soc. 118 (1965), 498-525. Google Scholar

[2] 2. Gillman, L. and Jerison, M., Rings of continuous functions, Van Nostrand, Princeton, N.J., 1960. Google Scholar

[3] 3. Henriksen, M., On the equivalence of the ring, lattice, and semi-group of continuous functions, Proc. Amer. Math. Soc. 7 (1956), 959-960. Google Scholar

[4] 4. Kist, J., Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. 13 (1963), 31-50. Google Scholar

[5] 5. Shirota, T., A generalization of a theorem of I. Kaplansky, Osaka J. Math. 4 (1952), 121-132. Google Scholar

[6] 6. Subramanian, H., Kaplansky's theorem for f-rings, Math. Ann. 179 (1968), 70-73. Google Scholar

[7] 7. Subramanian, H., Integer-valued continuous functions, Bull. Soc. Math. France 97 (1969). Google Scholar