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I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 1: Modules, Vector Spaces and Algebras ... ...
I need help with an aspect of Theorem 1.1 part 4 ...
Theorem 1.1 in Blyth reads as follows:View attachment 5838In the above text, in part 4 of the Theorem we read:" ... ... when $$R$$ is a division ring
(4) $$\lambda x = 0_M$$ implies $$\lambda = 0_R$$ or $$x = 0_M$$ ... ... "
Blyth proves that if $$R$$ is a division ring and $$\lambda x = 0_M$$ with $$\lambda \neq 0_R$$ then we have that $$x = 0_M$$ ... ...But ... ... Blyth does not show that if $$R$$ is a division ring and $$\lambda x = 0_M$$ with $$x \neq 0_M$$ then we have that $$\lambda = 0_R$$ ... ...Can someone please help me to prove this ...
Peter
I need help with an aspect of Theorem 1.1 part 4 ...
Theorem 1.1 in Blyth reads as follows:View attachment 5838In the above text, in part 4 of the Theorem we read:" ... ... when $$R$$ is a division ring
(4) $$\lambda x = 0_M$$ implies $$\lambda = 0_R$$ or $$x = 0_M$$ ... ... "
Blyth proves that if $$R$$ is a division ring and $$\lambda x = 0_M$$ with $$\lambda \neq 0_R$$ then we have that $$x = 0_M$$ ... ...But ... ... Blyth does not show that if $$R$$ is a division ring and $$\lambda x = 0_M$$ with $$x \neq 0_M$$ then we have that $$\lambda = 0_R$$ ... ...Can someone please help me to prove this ...
Peter
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