Associated prime ideals and regular elements in polynomial rings — New blog post on Freedom Math Dance

I discuss the theory of associated primes of a commutative ring, the description of zero divisors in polynomial rings, and the relation between these two things.

In some sense, this post is here to record a fact that ought to be better known and for which I ought to have known a proper proof beforehand.

https://freedommathdance.blogspot.com/2026/02/associated-prime-ideals-and-regular.html

#math #CommutativeAlgebra

Are there more people from algebraic geometry, commutative algebra, etc. on mastodon? Somehow I'm mostly following type theory people (which I'm also interested in, even though I'm even more of an amateur in type theory than in AG 😅)

#AlgebraicGeometry #CommutativeAlgebra

Anyone know if there is a survey of Gröbner(-Shirshov) bases out there that shows e.g. a big Venn diagram or containment diagram of which results generalize which other results? So many results on more and more general types of rings, I can't keep them all straight.

#algebra #ComputationalAlgebra #algorithms #math #CommutativeAlgebra

Abelian Artinian Noetherian Cohen-Macaulay Hilbert Jacobson rings

#algebra #CommutativeAlgebra

New blog post on Freedom Math Dance:
Flatness and projectivity: when is the localization of a ring a projective module?

Projective modules and flat modules are two important concepts in algebra, because they characterize those modules for which a general functorial construction (Hom module and tensor product, respectively) behave better than what is the case for general modules.

This blog post came out of reading a confusion on a student's exam: projective modules are flat, but not all flat modules are projective. Since localization gives flat modules, it is easy to obtain a an example of a flat module which is not projective (see below, Q works, as a Z-module), but my question was to understand when the localization of a commutative ring is a projective module.

Follow the link to read →

https://freedommathdance.blogspot.com/2024/04/flatness-and-projectivity-when-the.html

#CommutativeAlgebra #Math