There are three Athens Algebra Seminar features the week of March 26.
The March 27 seminar, from 4:30-5:30 p.m. in Morton 313, is a new installment the ongoing reading seminar on Nearrings. Doctoral student Majed Zalaiee is taking a reading course on the subject, and his lectures are part of the evaluation.
The March 29 seminar, from 4:30-5:30 p.m. in Morton 313, will be continuation of last week’s presentation by Lizbeth Sandoval, from UNAM, Mexico on “Topological spaces associated to a module.” She is a Fulbright scholar visiting the Center of Ring Theory and its Applications for one year.
On March 30, Sandoval will also be the speaker in Columbus at Ohio State University. The title and abstract are as follows:
Topological spaces associated to a module
Abstract: A topological space is it said to be {\it spectral} if it satisfies that is $T_0,$ quasicompact, has a basis of compact open subsets which is closed under finite intersection, and all irreducible closed subsets are closures of points (i.e. sober). In {\it Prime ideal structure in commutative rings,Trans. Amer. Math. Soc. 142 (1969)} M. Hochster characterized spectral topological spaces showing that a topological space $X$ is spectral if and only if it is homeomorphic to $Spec(R)$ for some commutative ring $R.$
Inspired by that result, we are interesting in the behavior of a spectrum for a module $M.$ In \cite{meAM}, we studied a prime spectrum for a module throught some associated frames, and we gave a module counterpart of the well-known result that in a commutative ring the set of semiprime ideals, that is, radical ideals is a frame. In \cite{meAML}, we continue this work, we define semiprimitive submodules and we prove that they form a spatial frame canonically isomorphic to the topology of Max(M). Also, we study the soberness of a prime spectrum for $M$ and for the subspace Max(M) in terms of the point space of that frame.
The purpose of this talk is to present some of these results. This is a jointly work with M. Medina-Barcenas, L. Morales-Callejas and A. Zaldivar-Corichi.
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