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I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ...
I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...
I need help with Bland's proof of the Third Isomorphism Theorem for rings ...
Bland's Third Isomorphism Theorem for rings and its proof read as follows:https://www.physicsforums.com/attachments/7973
In the above proof by Bland we read the following:
" ... ... The mapping $$f \ : \ I_1 \rightarrow ( I_1 + I_2 ) / I_2$$ given by $$f(x) = x + I_2$$ is a well-defined ring epimorphism with kernel $$I_1 \cap I_2$$. ... ... "I cannot see how $$f$$ can be an epimorphism as it does not seem to be onto $$( I_1 + I_2 ) / I_2$$ ... ...
My reasoning (which I strongly suspect is faulty) is as follows:... ... The domain of $$f$$ is $$I_1$$, so $$x \in I_1$$ ...
Now there exists elements $$y \in I_1 + I_2$$ such that $$y \in I_2 \ \ ( y = 0 + y \text{ where } 0 \in I_1, y \in I_2) $$
For such $$y$$ there is a coset in $$( I_1 + I_2 ) / I_2$$ of the form $$y + I_2$$ that is not in the range of $$f$$ ...
... so $$f$$ is not an epimorphism ...
It seems certain to me that my reasoning is wrong somewhere ... can someone please point out the error(s) in my analysis above ...
Peter
I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...
I need help with Bland's proof of the Third Isomorphism Theorem for rings ...
Bland's Third Isomorphism Theorem for rings and its proof read as follows:https://www.physicsforums.com/attachments/7973
In the above proof by Bland we read the following:
" ... ... The mapping $$f \ : \ I_1 \rightarrow ( I_1 + I_2 ) / I_2$$ given by $$f(x) = x + I_2$$ is a well-defined ring epimorphism with kernel $$I_1 \cap I_2$$. ... ... "I cannot see how $$f$$ can be an epimorphism as it does not seem to be onto $$( I_1 + I_2 ) / I_2$$ ... ...
My reasoning (which I strongly suspect is faulty) is as follows:... ... The domain of $$f$$ is $$I_1$$, so $$x \in I_1$$ ...
Now there exists elements $$y \in I_1 + I_2$$ such that $$y \in I_2 \ \ ( y = 0 + y \text{ where } 0 \in I_1, y \in I_2) $$
For such $$y$$ there is a coset in $$( I_1 + I_2 ) / I_2$$ of the form $$y + I_2$$ that is not in the range of $$f$$ ...
... so $$f$$ is not an epimorphism ...
It seems certain to me that my reasoning is wrong somewhere ... can someone please point out the error(s) in my analysis above ...
Peter