Help on Bresar's Left & Right Multiplication Maps on Algebras, Lemma 1.24

[Math Amateur]

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: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

 

[jvicens]

Introduction to Noncommutative Algebra by Matej Bresar is a comprehensive book that covers various topics in noncommutative algebra. In Chapter 1, Bresar focuses on finite dimensional division algebras.

Lemma 1.24 is an important result in this chapter and provides a key step in proving the Wedderburn-Artin theorem. The lemma states:

Lemma 1.24: Let A be a central simple algebra. If A is finite dimensional, then A is a division algebra.

In order to understand the statement and proof of this lemma, you have raised some questions which I will address below.

Question 1:

In the statement of Lemma 1.24, A is assumed to be a central simple algebra. This means that A is a finite dimensional algebra with a center that coincides with the field of scalars. In other words, A is a central algebra and every element of A can be written as a linear combination of elements from the field of scalars. This implies that A is unital, since the field of scalars contains the identity element 1_A. Therefore, your assumption is correct.

Question 2:

In the proof of Lemma 1.24, Bresar assumes that b_n \ne 0. This assumption is made in order to apply the induction hypothesis. The induction hypothesis states that if a central simple algebra A is finite dimensional and has a nonzero element, then A is a division algebra. So, by assuming that b_n \ne 0, Bresar is able to use the induction hypothesis to prove that A is a division algebra.

Question 3:

In the proof of Lemma 1.24, Bresar defines c_i = \sum_{ j = 1 }^m w_j b_i z_j and shows that c_n = 1 for some w_j, z_j \in A. The reason why n \gt 1 is because if n = 1, then c_1 = w_1 b_1 z_1 = 1. This would imply that b_1 is invertible, which contradicts our assumption that b_1 = 0. Therefore, n \gt 1.

The relevance of c_n = 1 and n \gt 1 in the proof is that it allows us to conclude that all the b_i = 0. This is because if c_n = 1 and n \gt 1, then we can write