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Let $R$ be a unitary commutative ring and $I$ an ideal in $R$. An element $f\in R$ is \emph{integral} over $I$, if there exists an equation
$$f^k+c_1f^{k-1}+\cdots +c_{k-1}f+c_k=0 ~~\mathrm{with} ~~ c_i\in I^i.$$
The set of elements $\overline{I}$ in $R$ which are integral over $I$ is the
\emph{integral closure} of $I$. The ideal $I$ is
\emph{integrally closed}, if $I=\overline{I}$, and $I$ is
\emph{normal} if all powers of $I$ are integrally closed.
This notion is linked to the graded algebras arising
from $I$ such as the Rees algebra $\mathrm{Rees}(I)=\oplus _{i\geq 0}I^{i}t^{i}$. It
is known that if $I$ is an ideal of a normal domain $R$, then $I$ is normal if and only if $\mathrm{Rees}(I)$ is normal. This brings up an importance of studying normality of ideals.
When $I$ is a monomial ideal in a
polynomial ring $R$, then $\overline{I}$ is the monomial ideal generated by all monomials $u \in R$ for which there exists an integer $k$ such that $u^{k}\in I^{k}$. In addition, it is well-known that every square-free monomial ideal is integrally closed.
Appearing as edge
and cover ideals of graphs, the square-free monomial ideals play a key role in connecting commutative algebra and combinatorics.
In this talk, we introduce techniques for producing normal square-free monomial ideals from old
such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations.
Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.
This is based on a joint work with Ibrahim ~Al-Ayyoub (Oman and Jordan), Mehrdad ~Nasernejad (Iran), Leslie G. Roberts (Canada), and Veronica Crispin Qui$\mathrm{\tilde{n}}$onez (Sweden) which will appear in ``Mathematica Scandinavica''.
https://zoom.us/j/9299700405?pwd=VTM5OW53dWFkbjcxQXBBalhiNWc2dz09
Meeting ID: 929 970 0405
Passcode: 210406
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