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 some further help with the statement and proof of Lemma 1.24 ...

Lemma 1.24 reads as follows:

My further questions regarding Bresar's statement and proof of Lemma 1.24 are as follows:

Question 1

In the statement of Lemma 1.24 we read the following:

" ... ... Let ##A## be a central simple algebra. ... ... "

I am assuming that since ##A## is central, it is unital ... that is there exists ##1_A \in A## such that ##x.1 = 1.x = 1## for all ##x \in A## ... ... is that correct ... ?

Question 2

In the proof of Lemma 1.24 we read the following:

" ... ... Suppose ##b_n \ne 0##. ... ... "

I am assuming that that the assumption ##b_n \ne 0## implies that we are also assuming

that ##b_1 = b_2 = \ ... \ ... \ = b_{n-1} = 0## ... ...

Is that correct?

Question 3

In the proof of Lemma 1.24 we read the following:

" ... ...where ##c_i = \sum_{ j = 1 }^m w_j b_i z_j## ; thus ##c_n = 1## for some ##w_j, z_j \in A## ... ...

This clearly implies that ##n \gt 1##. ... ... "

My question is ... why/how exactly must ##n \gt 1## ... ?

Further ... and even more puzzling ... what is the relevance to the proof of the statements that ##c_n = 1## and ##n \gt 1## ... ?

Why do we need these findings to establish that all the ##b_i = 0## ... ?

Hope someone can help ...

Peter

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*** NOTE ***

So that readers of the above post will be able to understand the context and notation of the post ... I am providing Bresar's first two pages on Multiplication Algebras ... ... as follows:

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# I Multiplication Maps on Algebras ... Further questions ...

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