Expand all Collapse all | Results 1 - 5 of 5 |
1. CMB 2010 (vol 53 pp. 667)
On the Endomorphism Rings of Local Cohomology Modules Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal
of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also
prove that, for a nonnegative integer $n$ such that
$H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for
all $i >0$ and $z \in \mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic
image of $R$, where $R_z$ is the ring of fractions of $R$ with
respect to a multiplicatively closed subset $\{z^j \mid j \geqslant
0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z
\in \mathfrak{a}$,
then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$
is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.
Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence Categories:13D45, 13D07, 13D25 |
2. CMB 2010 (vol 53 pp. 223)
Density of Polynomial Maps Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords:density, polynomial, endomorphism ring, PI Categories:16D60, 16S50 |
3. CMB 2007 (vol 50 pp. 632)
Transformations and Colorings of Groups Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.
Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion, Categories:05D10, 20D60, 22A10 |
4. CMB 2006 (vol 49 pp. 371)
Inner $E_0$-Semigroups on Infinite Factors This paper is concerned with the structure of
inner $E_0$-semigroups. We show that any inner
$E_0$-semigroup acting on an infinite factor
$M$ is completely determined by a continuous
tensor product system of Hilbert spaces in
$M$ and that the product system associated
with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.
Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacy Categories:46L40, 46L55 |
5. CMB 2006 (vol 49 pp. 265)
Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in
C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$-clean if every element is
the sum of a unit and a root of $g(x)$. If $V$ is a vector space of
countable dimension over a division ring $D,$ they showed that
$\end {}_{D}V$ is
$g(x)$-clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=x-x^{2}$
this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan.
In this paper we remove the countable condition, and in fact prove that
$\Mend {}_{R}M$ is $g(x)$-clean for any semisimple module $M$ over an arbitrary
ring $R$ provided that $g(x)\in (x-a)(x-b)C[x]$ where $a,b\in C$ and both $b$
and $b-a$ are units in $R$.
Keywords:Clean rings, linear transformations, endomorphism rings Categories:16S50, 16E50 |