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I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...

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

Lemma 1.24 reads as follows:View attachment 6258

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

In the statement of Lemma 1.24 we read the following:

" ... ... Let \(\displaystyle A\) be a central simple algebra. ... ... "I am assuming that since \(\displaystyle A\) is central, it is unital ... that is there exists \(\displaystyle 1_A \in A\) such that \(\displaystyle x.1_A = 1_A.x = 1\) for all \(\displaystyle x \in A\) ... ... is that correct ... ?

In the proof of Lemma 1.24 we read the following:

" ... ... Suppose \(\displaystyle b_n \ne 0\). ... ... "I am assuming that that the assumption \(\displaystyle b_n \ne 0\) implies that we are also assuming

that \(\displaystyle b_1 = b_2 = \ ... \ ... \ = b_{n-1} = 0\) ... ...

Is that correct?

In the proof of Lemma 1.24 we read the following:

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

This clearly implies that \(\displaystyle n \gt 1\). ... ... "My question is ... why/how exactly must \(\displaystyle n \gt 1\) ... ?

Further ... and even more puzzling ... what is the relevance to the proof of the statements that

\(\displaystyle c_n = 1\) and \(\displaystyle n \gt 1\) ... ?

Why do we need these findings to establish that all the \(\displaystyle b_i = 0\) ... ?Hope someone can help ...

Peter

===========================================================

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:View attachment 6259

View attachment 6260

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

Lemma 1.24 reads as follows:View attachment 6258

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 \(\displaystyle A\) be a central simple algebra. ... ... "I am assuming that since \(\displaystyle A\) is central, it is unital ... that is there exists \(\displaystyle 1_A \in A\) such that \(\displaystyle x.1_A = 1_A.x = 1\) for all \(\displaystyle x \in A\) ... ... is that correct ... ?

**Question 2**In the proof of Lemma 1.24 we read the following:

" ... ... Suppose \(\displaystyle b_n \ne 0\). ... ... "I am assuming that that the assumption \(\displaystyle b_n \ne 0\) implies that we are also assuming

that \(\displaystyle b_1 = b_2 = \ ... \ ... \ = b_{n-1} = 0\) ... ...

Is that correct?

**Question 3**In the proof of Lemma 1.24 we read the following:

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

This clearly implies that \(\displaystyle n \gt 1\). ... ... "My question is ... why/how exactly must \(\displaystyle n \gt 1\) ... ?

Further ... and even more puzzling ... what is the relevance to the proof of the statements that

\(\displaystyle c_n = 1\) and \(\displaystyle n \gt 1\) ... ?

Why do we need these findings to establish that all the \(\displaystyle b_i = 0\) ... ?Hope someone can help ...

Peter

===========================================================

***** 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:View attachment 6259

View attachment 6260

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