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1. CMB 2002 (vol 45 pp. 119)
The Grade Conjecture and the $S_{2}$ Condition 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.
Categories:13D22, 13D45, 13D25, 13C15 |
2. CMB 2000 (vol 43 pp. 312)
On the Prime Ideals in a Commutative Ring 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.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 |
3. CMB 1998 (vol 41 pp. 3)
Root closure in Integral Domains, III {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$\}$).
Categories:13G05, 13F25, 13C15, 13F45, 13B99, 12D99 |