Talk:Annihilator (ring theory)

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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

${\begin{aligned}V({\text{Ann}}_{R}(R/I))&=V((0))\cap V(I)\\&=V(I)\end{aligned}}$

Hence the annihilator of $R/I$ is just $I$. 70.171.155.43 (talk) 20:46, 30 January 2021 (UTC)