I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I need help with some aspects of Bresar's Example 1.10 on a simple matrix ring over a division ring ...

Example 1.10, including some preamble, reads as follows:

In the above text from Bresar we read the following:

" ... and hence also ##(d a_{jk}^{-1} ) E_{ii} \cdot a_{jk} E_{il} = d E_{il} \in I ## for every ##d \in D##. Consequently, ##I = M_n(D)##. ... ... "

My questions are as follows:

Question 1

I am assuming that ##(d a_{jk}^{-1} ) E_{ii} \cdot a_{jk} E_{il} = d E_{il}## because you can take the "scalars" out of the multiplication and multiply them as in

##c_1 (a_{ij} ) \cdot c_2 (b_{ij} ) = c_1 c_2 (a_{ij} ) \cdot (b_{ij} )##

Is that correct?

(Note: why we are messing with multiplications by scalars in a problem on rings, I don't know ... we seem to be treating the ring ##M_n (D)## as an algebra over ##D## ... )

Question 2

Bresar seems to be assuming that ##d E_{il} \in I ## for all ##1 \le i, l \le n## and for every ##d \in D## implies that ##I = M_n (D)## ...

But ... ... why exactly is this true ...

My thoughts ... maybe it is true because the ##E_{il}## generate the ring ##M_n (D)## ... or to put it another way ... any element in ##I## or ##M_n (D)## can be written uniquely in the form ##\sum_{i, j = 1}^n d_{ij} E_{ij}## ...and further, that all the ##E_{ij}## belong to I ...

Help with these questions will be appreciated ...

Peter

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So that readers of the above post can appreciate the relevant context of the post, I am providing the introduction to Section 1.3 Simple Rings ... as follows:

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# I Simple Matrix Rings ...

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