A Commutative Banach Algebra Which Factorizes But Has No Approximate Units

Elements in a Commutative Banach Algebra Determining the Norm Topology

For an element a of a commutative complex Banach algebra (A, ‖ · ‖) we investigate the following property: every complete norm | · | on A making the multiplication by a from (A, | · |) to itself continuous is equivalent to ‖ · ‖.

Commutative Algebra Commutative Algebra : General

A Model in Which Every Boolean Algebra Has Many Subalgebras

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subsetA such that 2|A| = 2|B|. This implies in particular that B has 2|B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from...

A GRAPH WHICH RECOGNIZES IDEMPOTENTS OF A COMMUTATIVE RING

In this paper we introduce and study a graph on the set of ideals of a commutative ring $R$. The vertices of this graph are non-trivial ideals of $R$ and two distinct ideals $I$ and $J$ are adjacent if and only $IJ=Icap J$. We obtain some properties of this graph and study its relation to the structure of $R$.

Commutative Algebra

Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2...