Sufficient conditions are given in order to prove the lowest unknown case of the grade conjecture. The proof combines vanishing results of local cohomology and the $S_{2}$ condition.

If $n$ and $m$ are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring $R$ with exactly $n$ elements and exactly $m$ prime ideals. Next, assuming the Axiom of Choice, it is proved that if $R$ is a commutative ring and $T$ is a commutative $R$-algebra which is generated by a set $I$, then each chain of prime ideals of $T$ lying over the same prime ideal of $R$ has at most $2^{|I|}$ elements. A polynomial ring example shows that the preceding result is best-possible.

{If A is a subring of a commutative ring B and if n is a positive integer, a number of sufficient conditions are given for ``A[[X]]is n-root closed in B[[X]]'' to be equivalent to ``A is n-root closed in B.'' In addition, it is shown that if S is a multiplicative submonoid of the positive integers ${\bbd P}$ which is generated by primes, then there exists a one-dimensional quasilocal integral domain A (resp., a von Neumann regular ring A) such that $S = \{ n \in {\bbd P}\mid A$ is $n$-root closed$\}$ (resp., $S = \{n \in {\bbd P}\mid A[[X]]$ is $n$-rootclosed$\}$).