Math Amateur
Gold Member
MHB
- 3,920
- 48
I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...
I am currently focused on Section 2.3: Artinian Rings: The Semisimple Case
I need help with some comments made by Cohn in the introduction to Section 2.3 ...
The relevant comments by Cohn read as follows:https://www.physicsforums.com/attachments/4965In the above text, Cohn writes the following:
" ... ... It follows that every cyclic right $$R$$-module is Artinian, for any such module is of the form $$R/A$$ for some right ideal $$A$$. ... "
I do not understand how/why the above statement follows ... can someone help ... ?
In particular, why is any cyclic right $$R$$-module of the form $$R/A$$ for some right ideal $$A$$ ... and further, why does this fact make any cyclic right $$R$$-module Artinian ... ??
Hope someone can help ... ...
Peter
I am currently focused on Section 2.3: Artinian Rings: The Semisimple Case
I need help with some comments made by Cohn in the introduction to Section 2.3 ...
The relevant comments by Cohn read as follows:https://www.physicsforums.com/attachments/4965In the above text, Cohn writes the following:
" ... ... It follows that every cyclic right $$R$$-module is Artinian, for any such module is of the form $$R/A$$ for some right ideal $$A$$. ... "
I do not understand how/why the above statement follows ... can someone help ... ?
In particular, why is any cyclic right $$R$$-module of the form $$R/A$$ for some right ideal $$A$$ ... and further, why does this fact make any cyclic right $$R$$-module Artinian ... ??
Hope someone can help ... ...
Peter