Explore BrainMass
Share

Simple Ring: Artenian Ring

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Please help with the following problem.

A simple ring is an Artenian ring with no 2-sided ideals. Let K be a field. Let V be an n-dimensional vector space over K and let A = EndK(V), the ring of n x n matrices over K. The ring is clearly Artenian since it has finite dimension over K. We want to see that the 2-sided ideal are either (0) or A. Let W be a subspace of V. Then LW = {f 2 A | f (W) = 0} is clearly a left ideal of A. Let I be a left ideal, W = { x2 V | f (x) = 0 for all f 2 I}, so I  LW. Let e1, . . . , en be a basis
of V with e1, . . . , er a basis of W.
Show that if f(x) has zero e1 - coordinate for all f 2 I, then x2 W. Note that if  permutes the ei, then   f 2 I.

© BrainMass Inc. brainmass.com March 21, 2019, 11:36 pm ad1c9bdddf
https://brainmass.com/math/ring-theory/simple-ring-artenian-ring-466001

Attachments

Solution Preview

Please see the attached file.

First note that we can identify the elements of the ring A with the linear transformations VV. Moreover, such linear transformations are uniquely determined by their images on e_1, e_2,...,e_n.
Let x∈V and f(x) has a zero ...

Solution Summary

This solution helps with a problem involving an Artenian ring. Step-by-step calculations are provided.

$2.19