Talk:Annihilator (ring theory)

Untitled
Annihilators in ring theory and linear algebra need separate treatments I think. Geometry guy 00:39, 22 May 2007 (UTC)

Confusing tag?
Please indicate which sections are the most confusing, thanks :) Rschwieb (talk) 01:41, 25 June 2011 (UTC)

Article improvements

 * Expand the definitions section to include the definition of annihilator for commutative rings
 * Also create subsections for left and right annihilators for noncommutative rings
 * Partition references by commutative and non-commutative references
 * Include references to noncommutative rings

Noncommutative properties

 * page 31 proposition 3.6 - http://math.uga.edu/~pete/noncommutativealgebra.pdf
 * starting page 81 - poset properties of left and right annihilators - https://pages.uoregon.edu/anderson/rings/COMPLETENOTES.PDF

Noncommutative examples

 * Include examples of annihilator for noncommutative rings
 * In matrix algebras, take a nilpotent matrix and find the annihilator of it
 * Include D-module examples: https://web.archive.org/web/20200513191733/http://cocoa.dima.unige.it/conference/cocoaviii/ucha.pdf

Additional references

 * A Term of Commutative Algebra - https://web.mit.edu/18.705/www/13Ed.pdf
 * NONCOMMUTATIVE RINGS - http://www-math.mit.edu/~etingof/artinnotes.pdf

this article is full of lies
what the heck happened here?!?! 70.171.155.43 (talk) 20:36, 30 January 2021 (UTC)

Here is the first lie excised from the article:


 * The prototypical example for an annihilator over a commutative ring can be understood by taking the quotient ring $$R/I$$ and considering it as a $$R$$-module. Then, the annihilator of $$R/I$$ is the ideal $$I$$ since all of the $$i \in I$$ act via the zero map on $$R/I$$. This shows how the ideal $$I$$ can be thought of as the set of torsion elements in the base ring $$R$$ for the module $$R/I$$. Also, notice that any element $$r \in R$$ that isn't in $$I$$ will have a non-zero action on the module $$R/I$$, implying the set $$R-I$$ can be thought of as the set of orthogonal elements to the ideal $$I$$. — Preceding unsigned comment added by 70.171.155.43 (talk) 20:41, 30 January 2021 (UTC)

This is the second one, a false proof of the first:


 * In particular, if $$M = R$$ then the annihilator of $$R/I$$ can be found explicitly using $$\begin{align}

V(\text{Ann}_R(R/I)) &= V((0)) \cap V(I) \\ &= V(I) \end{align}$$ Hence the annihilator of $$R/I$$ is just $$I$$. 70.171.155.43 (talk) 20:46, 30 January 2021 (UTC)