The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Togethe...

The purpose of this note is to record Hensel’s Lemma and a few of its immediate applications for my Math 831 class. At the end of the 19th century Hensel introduced the notion of completion into a number theoretic context in order to bring to bear techniques of analysis on some purely algebraic problems in number theory. One of his most crucial discoveries (in modern terms) was that when a ring...

A C *-algebra A is called an ideal C * -algebra (or equally a dual algebra) if it is an ideal in its bidual A**. M.C.F. Berglund proved that subalgebras and quotients of ideal C*-algebras are also ideal C*-algebras, that a commutative C *-algebra A is an ideal C *-algebra if and only if it is isomorphicto C (Q) for some discrete space ?. We investigate ideal J*-algebras and show that the a...

We examine primitive roots modulo the Fermat number Fm = 22 m + 1. We show that an odd integer n 3 is a Fermat prime if and only if the set of primitive roots modulo n is equal to the set of quadratic non-residues modulo n. This result is extended to primitive roots modulo twice a Fermat number.

This report gives an overview of the main ideas in Chapter 9 of the studied book, about the dimension of a variety. After recalling some definitions and basic properties about projective varieties, homogeneous ideals and Gröbner bases, we will begin our study of the dimension of a variety with the observation of a geometric way to define the dimension of a variety defined by a monomial ideal. T...

A. Grothendieck first coined the term “anabelian geometry” in a letter to G. Faltings [Gro97a] as a response to Faltings’ proof of the Mordell conjecture and in his celebrated Esquisse d’un Programme [Gro97b]. The “yoga” of Grothendieck’s anabelian geometry is that if the étale fundamental group πét 1 pX,xq of a variety X at a geometric point x is rich enough, then it should encode much of the ...

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n × r matrices as well as quantized factor algebras of M q (n) are analyzed. The latter are the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all (r + 1) × (r + 1) quantum subdeterminants and a certain localization of this algebra...