A COVERING CONDITION FOR THE PRIME SPECTRUMS

Let R be a commutative ring with identity, and let $f,\;g_i(i=1,\;\ldots,\;n),\;g_{\alpha}(\alpha{\in}S)$ be elements of R. We show that the following statements are equivalent; (i) $X_f{\subseteq}{\cup}_{\alpha{\in}S}X_{g\alpha}$ only if $X_f{\subseteq}X_{g\alpha}$ for some $\alpha{\in}S$, (ii) $V(f){\subseteq}{\cup}_{\alpha{\in}S}V(g_{\alpha})$ only if $V(f){\subseteq}V(g_{\alpha})$ for some $\alpha{\in}S$, (iii) $V(f){\subseteq}{\cup}^n_{i=1}V(g_i)$ only if $V(f){\subseteq}V(g_i)$ for some i, (iv) Spec(R) is linearly ordered under inclusion.