The Department of Mathematics at Western Michigan University presents an algebra seminar.

**Day and time:** Wednesdays from 11 a.m.-noon

**Place:** 6625 Everett Tower

The purpose of the Algebra Seminar is to reflect the research and scholarly interest of the Algebra Faculty: Clifton E. Ealy Jr – Quasi-groups, Groups, and related structures; Gene Freudenburg–Algebraic Geometry; Terrell Hodge – Algebraic groups and modular representation theory; John Martino–Group theory and classifying spaces of finite and compact groups; Annegret Paul–Representation of Lie groups; David Richter–Lie Algebras and matroids; and Jay Wood–coding theory over rings. The Algebra Seminar may be registered for as MATH 6930. The prerequisites for the Seminar as a course is graduate standing.

## upcoming events

**Oct. 2**

**Introduction to UFDs: Part 1** presented by Gene Freudenburg, Ph.D., Department of Mathematics, Western Michigan University

Abstract: The theme of the seminar this semester is: Unique Factorization Domains "with a View Toward Algebraic Geometry".

The first two lectures will lay out many of the main algebraic preliminaries for our investigations.

## Past events

**Sept. 25**

**Retracts of polynomial rings and the Abhyankar-Sathaye conjecture** presented by Takanori Nagamine, Ph.D., Niigata University (Japan)

Abstract: Let B = k[x_{1},...,x_{n}] be the polynomial ring in n ≥ 1 variables over a field k. A polynomial f in B is called a coordinate if there are g_{2},...,g_{n} in B such that k[f, g_{2},...,g_{n}] = B. The statement of Abhyankar-Sathaye conjecture is the following:

If B/f B ≈_{k} k[y_{1},...,y_{n-1}], then f is a coordinate.

In this talk, we will discuss this conjecture in terms of retracts of polynomial rings. In particular, we consider the case where n = 3 and k = C.

**Sept. 18**

**Closed polynomials over an integral domain: Part 2** presented by Hideo Kojima, Ph.D., Niigata University (Japan)

Abstract: In the second part, we will discuss closed polynomials over integral domains. Let R[X] be a polynomial ring in n variables over an integral domain R and let f in R[X]be a non-constant polynomial such that Q(R)[f] ∩ R[X] = R[f]. We give necessary and sufficient conditions for f to be closed in R[X], which generalize some results given in the first part. We also introduce some results of T. Nagamine and discuss some open questions.

**Sept. 11**

**Closed polynomials over an integral domain: Part 1** presented by Hideo Kojima, Ph.D., Niigata University (Japan)

Abstract: Let R[X] be a polynomial ring in n variables over an integral domain R. A non-constant polynomial f in R[X] is said to be closed if R[f] is integrally closed in R[X]. When R is a filed, we have several characterizations for a non-constant polynomial in R[X] to be closed. In these two talks, we will discuss closed polynomials over integral domains. First of all, we recall some results on closed polynomials over a field. Then we give some results on closed polynomials over a UFD.

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[g], where g in R[X] is a non-constant polynomial. The main result of this talk is that, for a non-constant polynomial F in R[X], R[f] is a maximal element of M(R, n) if and only if f is closed and Q(R)[f] ∩ R[X] = R[f], where Q(R) is the quotient field of R. Moreover, we give that, in the case where the characteristic of R equals zero, R[f] is a maximal elemenet of M(R, n) if and only if there exists an R-derivation on R[X] whose kernel equals R[f].