Let $K$ be a field, $A=K[X_1,\cdots ,X_n]$ and $𝕄$ the set of monomials of $A$. It is well known that the set of monomial ideals of $A$ is in a bijective correspondence with the set of all subsemiflows of the $𝕄$-semiflow $𝕄$. We generalize this to the case of term ideals of $A=R[X_1,\cdots ,X_n]$, where $R$ is a commutative Noetherian ring. A term ideal of $A$ is an ideal of $A$ generated by a family of terms $c{X}_1^{{\mu }_1}\cdots X_n^{{\mu }_n}$, where $c\in R$ and ${\mu }_1,\cdots ,{\mu }_n$ are integers $\ge 0$.

Let $R$ be a commutative ring with identity. If a ring $R$ is contained in an arbitrary union of rings, then $R$ is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in $R$, then $R$ contains one of them under various conditions.