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Homomorphisms from \(C(X,\mathbb {Z})\) into a ring of continuous functions. (English) Zbl 1398.54033

The system ZFC is assumed. Let \(X\) be a zero-dimensional Hausdorff space. Then \(X\) is called \(\mathbb{N}\)-compact if there exists a cardinal \(\kappa\) such that \(X\) is homeomorphic to a closed subspace of \(\mathbb{N}^{\kappa}\). It is known that there exists an \(\mathbb{N}\)-compact space \(v_0 X\) such that \(X\) is a dense subspace of \(v_0 X\) and each function \(f\in C(X, \mathbb{Z})\) has a continuous extension \(f^{v_0}\in C(v_0 X, \mathbb{Z})\). Let us recall that the Banaschewski compactification \(\beta_0 X\) of \(X\) is the Stone space of the Boolean algebra of clopen subsets of \(X\).
In the article, it is proved that if \(\Phi: C(X,\mathbb{Z})\to\mathbb{R}\) is a non-zero ring homomorphism, then \(\Phi(C(X,\mathbb{Z}))=\mathbb{Z}\) and there exists a unique point \(x\in v_0 X\) such that \(\Phi(f)=f^{v_0}(x)\) for each \(f\in C(X,\mathbb{Z})\). It is deduced that if \(Y\) is a Tychonoff space, while \(\Phi: C(X, \mathbb{Z})\to C(Y)\) is a non-zero ring homomorphism, then there exists a continuous mapping \(\pi:Y\to v_0 X\) such that \(\Phi(f)=f^{v_0}\circ \pi\) for each \(f\in C(X,\mathbb{Z})\) and, moreover, there exists a subset \(A\) of \(v_0 X\) such that \(Ker(\Phi)=\bigcap\{P^{a}: a\in A\}\) where \(P^{a}=\{f\in C(X, \mathbb{Z}): f^{v_0}(a)=0\}\) is a minimal prime ideal of \(C(X, \mathbb{Z})\).
If \(\Phi: C(X,\mathbb{Z})\to\mathbb{R}\) is a non-zero ring homomorphism, then there exists a unique point \(a\in v_0 X\) such that \(Ker(\Phi)=P^{a}\). If \(R\) is a subring of \(\mathbb{R}\), then the collection of all functions \(f\in C(X, R)\) such that the set \(\{x\in X: f(x)\neq 0\}\) is finite is denoted by \(C_F(X, R)\). The collection of all locally constant functions from \(C(X, \mathbb{Q})\) is denoted by \(C_{lc}(X, \mathbb{Q})\). The classical ring of quotients of a ring \(R\) is denoted by \(Q_{cl}(R)\).
It is shown that the rings \(Q_{cl}(C(X, \mathbb{Z}))\) and \(C_{lc}(X, \mathbb{Q})\) are isomorphic. Moreover, \(C_{lc}(X, \mathbb{Q})\) is the unique von Neumann regular subring of \(C(X,\mathbb{Q})\) which contains \(C(X, \mathbb{Z})\). That \(C(X, \mathbb{Q})\) is von Neumann regular is equivalent to any of the following conditions: (i) \(X\) is a \(P\)-space, (ii) \(C(X, \mathbb{Q})\) is \(\aleph_0\)-self-injective, (iii) \(Q_{cl}(C(X, \mathbb{Z}))\) is \(\aleph_0\)-self-injective; (iv) \(Q_{cl}(C(X, \mathbb{Z}))\) is isomorphic with \(C(Y, \mathbb{Q})\) for some zero-dimensional space \(Y\).
The space \(X\) is an extremally disconnected \(P\)-space if and only if \(C(X, \mathbb{Z})\) is an \(I\)-ring. Other relevant results and some properties of the factor ring \(\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})}\) are also given. Among them, it is shown that \(\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})}\) can be embedded in a ring of continuous functions if and only if each infinite subset of isolated points of \(X\) has a limit point in \(v_0 X\). In consequence, the factor ring \(\frac{\prod_{x\in X}\mathbb{Z}_x}{\bigoplus_{x\in X}\mathbb{Z}_x}\) is not embedded in any ring of continuous functions when \(X\) is infinite and \(\mathbb{Z}_x=\mathbb{Z}\) for each \(x\in X\). If \(X\) is a Stone space, then \(\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})}\) is isomorphic to \(C(X\setminus\mathbb{I}(X), \mathbb{Z})\) where \(\mathbb{I}(X)\) stands for the set of all isolated points of \(X\). The ring \((\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})})^{\ast}\) of all bounded elements of \(\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})}\) is isomorphic to \(C(\beta_0 X\setminus \mathbb{I}(X), \mathbb{Z})\). The classical ring of quotients of \(\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})}\) is isomorphic to the factor ring \(\frac{C_{lc}(X,\mathbb{Q})}{C_F(X,\mathbb{Q})}\). If both \(X\) and \(Y\) are zero-dimensional, then \(v_0 X\) and \(v_0 Y\) are homeomorphic if and only if \(C(X, \mathbb{Z})\) and \(C(Y, \mathbb{Z})\) are isomorphic, while \(\beta_0 X\setminus \mathbb{I}(X))\) and \(\beta_o Y\setminus\mathbb{I}(Y)\) are homeomorphic if and only if \((\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})})^{\ast}\) and \((\frac{C(Y,\mathbb{Z})}{C_F(Y,\mathbb{Z})})^{\ast}\) are isomorphic. Finally, it is proved that \(\frac{C(X,\mathbb{Z})}{C_F(X,\mathbb{Z})}\) is an \(I\)-ring if and only if each infinite subset of isolated points of \(X\) has a limit point in \(v_0 X\) and \(v_0 X\setminus\mathbb{I}(X)\) is an extremally disconnected \(P\)-space which is \(C_{\mathbb{Z}}\)-embedded in \(v_0 X\). Among other references, the article [F. M. Vechtomov, J. Math. Sci. 78, 702–753 (1996; Zbl 0868.46018)] is very helpful.

MSC:

54C40 Algebraic properties of function spaces in general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H13 Topological fields, rings, etc. (topological aspects)
54C05 Continuous maps
54C30 Real-valued functions in general topology

Citations:

Zbl 0868.46018
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References:

[1] Alling, NL, Rings of continuous integer-valued functions and nonstandard arithmetic, Trans. Am. Math. Soc., 118, 498-525, (1974) · Zbl 0143.01002 · doi:10.1090/S0002-9947-1965-0184960-6
[2] Azarpanah, F, Algebraic properties of some compact spaces, Real Anal. Exch., 25, 317-328, (2000) · Zbl 1015.54008
[3] Călugăreanu, G.: Lattice Concepts of Module Theory. Springer Science+Business Media, Dordrecht (2000) · Zbl 0959.06001
[4] Drees, KM, A Nagata-like theorem for certain function spaces, Algebra Universalis, 62, 259-272, (2009) · Zbl 1205.54025 · doi:10.1007/s00012-010-0050-y
[5] Eggert, N, Rings whose overrings are integrally closed, J. Reine Angew. Math., 282, 88-95, (1976) · Zbl 0318.13008
[6] Ellis, R, Extending continuous functions on zero-dimensional spaces, Math. Ann., 186, 114-122, (1970) · Zbl 0182.25501 · doi:10.1007/BF01350686
[7] Engelking, R; Mrówka, S, On E-compact spaces, Bull. Acad. Pol. Sci., 6, 429-436, (1958) · Zbl 0083.17402
[8] Estaji, AA; Karamzadeh, OAS, On \(C(X)\) modulo its socle, Commun. Algebra, 31, 1561-1571, (2003) · Zbl 1025.54012 · doi:10.1081/AGB-120018497
[9] Finn, RT; Martinez, J; McGovern, WW; Holland, WC (ed.); Martinez, J (ed.), Commutative singular \(f\)-rings, 149-166, (1997), Dordrecht · doi:10.1007/978-94-011-5640-0_6
[10] Ghadermazi, M; Karamzadeh, OAS; Namdari, M, On the functionally countable subalgebra of \(C(X)\), Rend. Sem. Mat. Univ. Padova, 129, 47-69, (2013) · Zbl 1279.54015 · doi:10.4171/RSMUP/129-4
[11] Gillman, L., Jerison, M.: Rings of Continuous Functions. Springer, Dordrecht (1976) · Zbl 0327.46040
[12] Hager, AW; Martinez, J, Fraction-dense algebras and spaces, Acta Appl. Math., 27, 55-65, (1992) · Zbl 0789.13005 · doi:10.1007/BF00046636
[13] Karamzadeh, OAS, On a question of matlis, Commun. Algebra, 25, 2717-2726, (1997) · Zbl 0878.16003 · doi:10.1080/00927879708826017
[14] Karamzadeh, OAS; Rostami, M, On the intrinsic topology and some related ideals of \(C(X)\), Proc. Am. Math. Soc., 93, 179-184, (1985) · Zbl 0524.54013
[15] Martinez, J, \(C(X, \mathbb{Z}),\) revisited, Adv. Math., 99, 152-161, (1993) · Zbl 0855.54021 · doi:10.1006/aima.1993.1022
[16] Martinez, J, The maximal ring of quotients of an \(f\)-ring, Algebra Universalis, 33, 355-369, (1995) · Zbl 0819.06018 · doi:10.1007/BF01190704
[17] McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley Interscience, New York (1987) · Zbl 0644.16008
[18] Mrówka, S, Structures of continuous functions III. rings and lattices of integer-valued continuous functions, Vehr. Ned. Akad. Wet. Sect. I, 68, 74-82, (1965) · Zbl 0139.07404
[19] Pierce, RS, Rings of integer-valued continuous functions, Trans. Am. Math. Soc., 100, 371-394, (1961) · Zbl 0196.15401 · doi:10.1090/S0002-9947-1961-0131438-8
[20] Porter, J.R., Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York (1988) · Zbl 0652.54016 · doi:10.1007/978-1-4612-3712-9
[21] Sharp, R.Y.: Steps in Commutative Algebra, 2nd edn. Cambridge University Press, Cambridge (2000) · Zbl 0969.13001
[22] Subramanian, H, Integer-valued continuous functions, Bull. Soc. Math. Fr., 95, 275-263, (1969) · Zbl 0186.35301 · doi:10.24033/bsmf.1681
[23] Vechtomov, EM, Rings of continuous functions with values in a topological division ring, J. Math. Sci., 78, 702-753, (1996) · Zbl 0868.46018 · doi:10.1007/BF02363066
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