Homomorphisms from $C(X,\mathbb {Z})$ into a ring of continuous functions

Abstract

Let X be a zero-dimensional space and Y be a Tychonoff space. We show that every non-zero ring homomorphism $\Phi :C(X,\mathbb {Z})\rightarrow C(Y)$ can be induced by a continuous function $\pi :Y\rightarrow \upsilon _0X.$ Using this, it turns out that the kernel of such homomorphisms is equal to the intersection of some family of minimal prime ideals in ${{\mathrm{MinMax}}}\left( C(X,\mathbb {Z})\right) .$ As a consequence, we are able to obtain the fact that the factor ring $\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}$ is a subring of some ring of continuous functions if and only if each infinite subset of isolated points of X has a limit point in $\upsilon _0X.$ This implies that for an arbitrary infinite set X,  the factor ring $\frac{\prod _{_{x\in X}}\mathbb {Z}_{_{x}}}{\oplus _{_{x\in X}}\mathbb {Z}_{_{x}}}$ is not embedded in any ring of continuous functions. The classical ring of quotients of the factor ring $\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}$ is fully characterized. Finally, it is shown that the factor ring $\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}$ is an I-ring if and only if each infinite subset of isolated points on X has a limit point in $\upsilon _0X$ and $\upsilon _0X{\setminus }\mathbb {I}(X)$ is an extremally disconnected $C_{\mathbb {Z}}$-subspace of $\upsilon _0X,$ where $\mathbb {I}(X)$ is the set of all isolated points of X.