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1. CMB Online first
On Stanley Depths of Certain Monomial Factor Algebras Let $S=K[x_1,\ldots,x_n]$
be the polynomial
ring in $n$-variables over a field $K$ and $I$ a monomial ideal
of $S$. According to one standard primary decomposition of $I$,
we get a Stanley decomposition of the monomial factor algebra
$S/I$.
Using this Stanley decomposition, one can estimate the Stanley
depth of $S/I$. It is proved that
${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {size}}_S(I)$. When $I$ is squarefree
and ${\operatorname {bigsize}}_S(I)\leq 2$, the Stanley conjecture holds
for
$S/I$, i.e., ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {depth}}_S(S/I)$.
Keywords:monomial ideal, size, Stanley depth Categories:13F20, 13C15 |
2. 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 |
3. 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 |
4. 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 |