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)[reply]

Please indicate which sections are the most confusing, thanks :) Rschwieb (talk) 01:41, 25 June 2011 (UTC)[reply]

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

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

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

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