Homomorphisms from $$C(X,\mathbb {Z})$$ C ( X , 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.