November 1, 2019 at 8:45 pm

Algebra Seminar | Pallone on Amenable and Simple bases for the Algebra of Entangled Polynomials, Nov. 14

Ashley Pallone, portrait

The Algebra Seminar series meets twice this week, with meetings on Tuesday, Nov. 12, and Thursday, Nov. 14, from 5 to 6 p.m. in Morton 313. In addition, a visiting undergraduate student from Eastern Kentucky University giving us a talk about his project on an innovative perspective on the Lasker-Noether Theorem on Wednesday, Nov. 13, from 4 to 5 p.m. in Morton 320.

The speaker on Tuesday is Aaron Nicely, a Mathematics teaching assistant, continuing the series of presentations about Left Braces. Left Braces are an algebraic structure that generalizes Jacobson radical rings and are useful to find solutions to the Yang-Baxter equations.

Wednesday Meeting

Jesse Keyes of Eastern Kentucky University presents “Primary Decomposition in Noetherian Rings.”

Abstract: The notion of primary decomposition will be explored in both commutative and non-commutative Noetherian rings. First, proof of the Lasker-Noether theorem will be given
demonstrating that every ideal in a commutative Noetherian ring has a primary decomposition. On the other hand, an example will be discussed showing that ideals in non-commutative Noetherian rings do not always have a primary decomposition.

Thursday Meeting

Thursday’s speaker is Ashley Pallone, a Mathematics graduate student at Ohio University, discussing “Amenable and Simple bases for the Algebra of Entangled Polynomials.”

Abstract: Using a standard embedding of K[x], the algebra of polynomials with coefficients in a K, into the ring of row-column-finite matrices over K, non-trivial factorizations of some irreducible polynomials in K[x] are possible. The row-column-finite matrices involved in those factorizations resemble polynomials in several ways. We call these matrices “entangled polynomials” as each one of them is induced by a finite number of polynomials. For every fixed m \ge 1, entangled polynomials induced by m polynomials (the so-called m-nomials) form a ring K^{(m)}[x]. The ring of 1-nomials is precisely K[x]. When m divides n, K^{(m)}[x] is a subring of K^{(n)}[x]. In particular, all rings of m-nomials include the polynomials.

Given an algebra A over a field K, a basis B for A is said to be amenable if one can naturally extend the A-module structure on the F-vector space F^(B) to the vector space F^B.

A basis B is congenial to another one C if infinite linear combinations of elements of B translate in a natural way to infinite linear combinations of elements of C. While congeniality is not symmetric in general, when two bases B and C are mutually congenial then B is amenable if and only if C is amenable and, in that case, the module structures obtained on K^B and K^C are isomorphic. An interesting feature of congeniality is that (not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce.

An amenable basis B is simple if it is not properly congenial to any other amenable basis.

We will present amenable bases and simple bases in the ring of all 2-nomials as well as some results about congeniality in that setting.

This is a report on ongoing work by the speaker and Dr. Sergio R. López-Permouth, Professor of Mathematics at Ohio University.

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