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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 \(\displaystyle I\) is an ideal of \(\displaystyle \mathbb{Z}\) such that \(\displaystyle 5 \mathbb{Z} \subset I \subseteq \mathbb{Z}\) then \(\displaystyle I = \mathbb{Z}\) ... Bland then claims that \(\displaystyle I\) is a maximal ideal of \(\displaystyle \mathbb{Z}\) ...... BUT ...... doesn't Bland also have to show that if \(\displaystyle I\) is an ideal of \(\displaystyle \mathbb{Z}\) such that \(\displaystyle 5 \mathbb{Z} \subseteq I \subset \mathbb{Z}\) then \(\displaystyle I = 5 \mathbb{Z}\) ... ?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 \(\displaystyle I\) is an ideal of \(\displaystyle \mathbb{Z}\) such that \(\displaystyle 5 \mathbb{Z} \subset I \subseteq \mathbb{Z}\) then \(\displaystyle I = \mathbb{Z}\) ... Bland then claims that \(\displaystyle I\) is a maximal ideal of \(\displaystyle \mathbb{Z}\) ...... BUT ...... doesn't Bland also have to show that if \(\displaystyle I\) is an ideal of \(\displaystyle \mathbb{Z}\) such that \(\displaystyle 5 \mathbb{Z} \subseteq I \subset \mathbb{Z}\) then \(\displaystyle I = 5 \mathbb{Z}\) ... ?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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