Bresar, Lemma 1.2 - Finite Division Algebras ....

[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 help with some aspects of the proof of Lemma 1.2 ... ...

Lemma 1.2 reads as follows:
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My questions related to the above proof by Bresar are as follows:Question 1

In the above text from Bresar we read the following:

" ... ... Since \(\displaystyle u, v, 1\) are linearly independent, this yields \(\displaystyle \lambda + \mu = \lambda - \mu = 0\), hence \(\displaystyle \lambda = \mu = 0\), and \(\displaystyle u + v \in V\) follows from the first paragraph. ... ... "My question is ... ... how exactly does it follow that \(\displaystyle u + v \in V\)?
Question 2

In the above text from Bresar we read the following:

" ... ... Again, using the observation from the first paragraph we see that \(\displaystyle x + \frac{v}{2} \in V\). Accordingly, \(\displaystyle x = - \frac{v}{2} + ( x + \frac{v}{2} ) \in \mathbb{R} \oplus V\). ... ... "My question is ... ... how exactly does it follow that \(\displaystyle x + \frac{v}{2} \in V\) and, further, how exactly does it then follow that \(\displaystyle x = - \frac{v}{2} + ( x + \frac{v}{2} ) \in \mathbb{R} \oplus V\) ... ... ?
Hope someone can help ...

Peter
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In order for readers of the above post to appreciate the context of the post I am providing pages 1-2 of Bresar ... as follows ...View attachment 6202
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[jvicens]

Pages 1-2 of Bresar's "Introduction to Noncommutative Algebra" provide background information on division algebras and their properties. In particular, Bresar defines a division algebra as a finite-dimensional associative algebra over a field that has a multiplicative identity and every nonzero element has a multiplicative inverse. He also introduces the concept of a simple algebra, which is a division algebra that has no proper nonzero two-sided ideals.

In Chapter 1, Bresar focuses on finite-dimensional division algebras over the real numbers. He starts by proving the Wedderburn-Artin theorem, which states that every finite-dimensional division algebra over the real numbers is isomorphic to either the real numbers, the complex numbers, or the quaternions. He then moves on to studying the properties of these three division algebras.

Lemma 1.2 is part of the proof of Proposition 1.4, which states that every finite-dimensional division algebra over the real numbers is isomorphic to one of the three mentioned above. In this lemma, Bresar shows that for any nonzero element x in a finite-dimensional division algebra over the real numbers, there exists a real number \lambda and an element v in the algebra such that x = \lambda + v. This is a key step in the proof of Proposition 1.4.

Now, let's address the two questions posed in the forum post.

Question 1:

In the proof of Lemma 1.2, Bresar shows that for any nonzero element x in the division algebra, there exists a real number \lambda and an element v such that x = \lambda + v. To show that u + v \in V, Bresar uses the fact that u, v, and 1 are linearly independent. This means that there is no nontrivial linear combination of these three elements that equals 0. So, if we assume that u + v \in V, then we would have \lambda + \mu = 0 and \lambda - \mu = 0, which leads to \lambda = \mu = 0. This contradicts the fact that u and v are linearly independent, so we must conclude that u + v \notin V.

Question 2:

In the second paragraph of the proof, Bresar shows that x + \frac{v}{2} \in V. To do this, he uses the observation from the first paragraph, which states that x