Does Proposition 3.2.6 Imply g'(y) = x' - f(f'(x'))?

In summary, the conversation is about the book "Rings and Their Modules" by Paul E. Bland, specifically focusing on Section 3.2 and Proposition 3.2.6. The speaker is asking for help in understanding the proof of the proposition and the response explains that the text implies $g'(y) = x' - f(f'(x'))$ and provides an intuitive explanation.
  • #1
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I am reading Paul E. Bland's book "Rings and Their Modules" ...

Currently I am focused on Section 3.2 Exact Sequences in \(\displaystyle \text{Mod}_R\) ... ...

I need some further help in order to fully understand the proof of Proposition 3.2.6 ...

Proposition 3.2.6 and its proof read as follows:
View attachment 8079
In the above proof of Proposition 3.2.6 we read the following:"... ... now define \(\displaystyle g' \ : \ M_2 \longrightarrow M\) by \(\displaystyle g'(y) = x - f(f'(x))\), where \(\displaystyle x \in M\) is such that \(\displaystyle g(x) = y\) ... ... ... ...

... ... Suppose that \(\displaystyle x' \in M\) is also such that \(\displaystyle g(x') = y\) ... ... Does the above text imply that \(\displaystyle g'(y) = x' - f( f'(x') )\) ... ... ?

Peter
 
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  • #2
Hi Peter,

Yes, indeed, $g'(y) = x-f(f'(x)) = x' - f(f'(x'))$, since the last equality has just been proved. This shows that $g'(y)$ in unambiguously defined; please refer to my previous post for an intuitive explanation.
 

FAQ: Does Proposition 3.2.6 Imply g'(y) = x' - f(f'(x'))?

What is Proposition 3.2.6 in "Split Short Exact Sequences" by Bland?

Proposition 3.2.6 is a mathematical statement in the book "Split Short Exact Sequences" by Bland. It states that in a split short exact sequence of vector spaces, the middle term is isomorphic to the direct sum of the first and last terms.

What are split short exact sequences?

Split short exact sequences are a type of sequence in mathematics that involves vector spaces and linear transformations. They are called "short" because they only have three terms, and "exact" because they satisfy certain conditions related to the kernels and images of the linear transformations.

What does Proposition 3.2.6 tell us about split short exact sequences?

Proposition 3.2.6 provides a useful property of split short exact sequences, stating that the middle term is isomorphic to the direct sum of the first and last terms. This can be helpful in understanding and solving problems involving split short exact sequences.

What is the significance of Proposition 3.2.6 in mathematics?

Proposition 3.2.6 is a useful tool in understanding and working with split short exact sequences, which have applications in various areas of mathematics, including algebra, topology, and geometry. It allows us to simplify and analyze these sequences in a more efficient way.

Are there any other important propositions or theorems related to split short exact sequences?

Yes, there are several other important propositions and theorems related to split short exact sequences, including the splitting lemma, the short five lemma, and the snake lemma. These all provide useful information and techniques for working with split short exact sequences.

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