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I am reading The Basics of Abstract Algebra by Paul E. Bland ...
I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ...
I need help with the proof of Example 2, Section 3.2.12, pages 147 to 148 ... ... Example 2, Section 3.2.12 reads as follows:
View attachment 8262
https://www.physicsforums.com/attachments/8263
In the above example Bland shows that if $$I$$ is an ideal of $$\mathbb{Z}$$ such that $$5 \mathbb{Z} \subset I \subseteq \mathbb{Z}$$ then $$I = \mathbb{Z}$$ ... Bland then claims that $$I$$ is a maximal ideal of $$\mathbb{Z}$$ ...... BUT ...... doesn't Bland also have to show that if $$I$$ is an ideal of $$\mathbb{Z}$$ such that $$5 \mathbb{Z} \subseteq I \subset \mathbb{Z}$$ then $$I = 5 \mathbb{Z}$$ ... ?Can someone explain why Bland's proof is complete as it stands ...
Peter============================================================================***NOTE***
It may help readers to have access to Bland's definition of a maximal ideal ... so I am providing the same as follows:https://www.physicsforums.com/attachments/8264
https://www.physicsforums.com/attachments/8265Sorry about the legibility ... but Bland shades his definitions ...Peter
I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ...
I need help with the proof of Example 2, Section 3.2.12, pages 147 to 148 ... ... Example 2, Section 3.2.12 reads as follows:
View attachment 8262
https://www.physicsforums.com/attachments/8263
In the above example Bland shows that if $$I$$ is an ideal of $$\mathbb{Z}$$ such that $$5 \mathbb{Z} \subset I \subseteq \mathbb{Z}$$ then $$I = \mathbb{Z}$$ ... Bland then claims that $$I$$ is a maximal ideal of $$\mathbb{Z}$$ ...... BUT ...... doesn't Bland also have to show that if $$I$$ is an ideal of $$\mathbb{Z}$$ such that $$5 \mathbb{Z} \subseteq I \subset \mathbb{Z}$$ then $$I = 5 \mathbb{Z}$$ ... ?Can someone explain why Bland's proof is complete as it stands ...
Peter============================================================================***NOTE***
It may help readers to have access to Bland's definition of a maximal ideal ... so I am providing the same as follows:https://www.physicsforums.com/attachments/8264
https://www.physicsforums.com/attachments/8265Sorry about the legibility ... but Bland shades his definitions ...Peter
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