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 the proof of Lemma 1.25 ...

Lemma 1.25 reads as follows:

My questions on the proof of Lemma 1.25 are as follows:

Question 1

In the above text from Bresar we read the following:

" ... ... Therefore ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]## ... ... "

Can someone please explain exactly why Bresar is concluding that ##[ M(A) \ : \ F ] \ge d^2## ... ... ?

Question 2

In the above text from Bresar we read the following:

" ... ... Therefore ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]##

and so ##M(A) = [ \text{ End}_F (A) \ : \ F ]##. ... ... "

Can someone please explain exactly why ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]## ... ...

... implies that ... ##M(A) = [ \text{ End}_F (A) \ : \ F ]## ...

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 ... Bresar, Lemma 1.25 ...

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