Category Archives: commutative-algebra

An Abelian Group of Endoprojective Dimension One

We already saw that an abelian group with a $\Z$-direct summand is projective over its endomorphism ring. Finitely generated abelian groups are also projective over their endomorphism rings by essentially the same argument. What's an example of an abelian group that is not projective over its endomorphism ring? Here's one: the multiplicative group $Z(p^\infty)$ of […]

Local Rings of Global Dimension at Most Two

Wolmer Vasconcelos [1] gave the following classification theorem about commutative local rings of global dimension two: Theorem. Let $A$ be a commutative local ring of global dimension two with maximal ideal $M$. If $M$ is principal or not finitely generated, then $A$ is a valuation domain. Otherwise $M$ is generated by a regular sequence of […]

Commutative von Neumann Regular Rings

A ring of left global dimension zero is a ring $R$ for which every left $R$-module is projective. These are also known as semisimple rings of the Wedderburn-Artin theory fame, which says that these rings are precisely the finite direct products of full matrix rings over division rings. Note the subtle detail that "semisimple" is […]

Book Review: deMeyer and Ingraham's "Separable Algebras over Commutative Rings"

Let $R$ be a commutative ring. We say that an $R$-algebra $A$ is separable if it is projective as an $A\otimes_R A^{\rm op}$-module. Examples include full matrix rings over $R$, finite separable field extensions, and $\Z[\tfrac 12,i]$ as a $\Z[\tfrac 12]$-algebra. The 1970 classic Separable Algebras by deMeyer and Ingraham acquaints the reader with this […]

Example: Separability Idempotent for a Field Extension

Let $R$ be a commutative ring and let $A$ be an $R$-algebra. We say that $A$ is separable if $A$ is projective as an $A\otimes_RA^{\rm op}$-module. There is a multiplication map $\mu:A\otimes_RA^{\rm op}\to A$ given by $a\otimes a'\mapsto aa'$, whose kernel we'll call $J$. It's a fact that $A$ is separable if and only if […]

Automorphisms of Matrix Rings over Fields are Inner

Let $k$ be a field. The ring $M_n(k)$ of $n\times n$ matrices over $k$ has some automorphisms, given by conjugation by elements of $\GL_n(k)$. These are inner automorphisms, and this action happens to be the adjoint action of $\GL_n$ on its Lie algebra. Are there any other automorphisms? The answer is no, and the reason […]

Yet Another Algebra that is not Separable

Let $R$ be a commutative ring and $A$ be an $R$-algebra. We say that $A$ is a separable $R$-algebra if $A$ is projective as an $A\otimes_R A^{\rm op}$-module, where the action of $A\otimes_RA^{\rm op}$ is given by $(a\otimes a')b = aba'$. We already showed that the ring of upper triangular matrices over a commutative ring […]