Understanding Lemma 4.3.12 in Paul E. Bland's Book: "Rings and Their Modules"

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In summary, In the book "Rings and Their Modules" by Paul E. Bland, Section 4.3 focuses on Modules Over Principal Ideal Domains. The reader asks for further help understanding the proof of Lemma 4.3.12, specifically where the special properties of a PID are used. Another reader provides the answer on page 120 of the book, stating that a greatest common divisor may fail to exist in a non-PID ring, but exists in a PID. The reader is grateful for the help.
  • #1
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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some further help in order to fully understand the proof of Lemma 4.3.12 ... ...

Lemma 4.3.12 reads as follows:View attachment 8316My question is as follows:

In the Lemma R is a PID ... where in the proof are the special properties of a PID used/needed ... it seems to me that the argument of the proof would hold valid for an ordinary commutative ring with identity ...

Can someone please point out the points in the proof where the special properties of a PID are needed ...

Peter

==============================================================================

It may help MHB
readers of the above post to have access to Bland's definition of a primitive element ... so I am providing the same as follows:View attachment 8317
Hope that helps ...

Peter
 
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The answer to your question is on page 120 of Bland, quote: "A greatest common divisor of a set of nonzero elements of R may fail to exist. The case is different for a principal ideal ring."
So, if you want existing gcd's, your ring has to be a PID.
 
  • #3
steenis said:
The answer to your question is on page 120 of Bland, quote: "A greatest common divisor of a set of nonzero elements of R may fail to exist. The case is different for a principal ideal ring."
So, if you want existing gcd's, your ring has to be a PID.
Thanks Steenis ...

I appreciate your help ...

Peter
 

1. What is Lemma 4.3.12 in Paul E. Bland's book "Rings and Their Modules"?

Lemma 4.3.12 in Paul E. Bland's book "Rings and Their Modules" is a mathematical statement that is used to prove certain properties of rings and their modules.

2. Why is Lemma 4.3.12 important in the study of rings and their modules?

Lemma 4.3.12 is important because it helps to establish important relationships and properties between rings and their modules, which are essential for understanding these mathematical structures.

3. What does Lemma 4.3.12 state?

Lemma 4.3.12 states that if a ring R is a direct product of two subrings A and B, then every R-module is isomorphic to a direct product of an A-module and a B-module.

4. How is Lemma 4.3.12 used in practice?

Lemma 4.3.12 is used in practice to simplify and prove certain properties of rings and their modules. It can also be used to prove other theorems and lemmas in the field of abstract algebra.

5. Are there any variations of Lemma 4.3.12?

Yes, there are variations of Lemma 4.3.12 that may have slightly different conditions or conclusions. However, they all serve the same purpose of establishing relationships between rings and their modules.

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