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) $X_f{subseteq}{cup}_{alpha{in}S}X_{galpha}$ only if $X_f{subseteq}X_{galpha}$ 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.