Principal Ideal Rings and GCDs .... .... Bland Proposition 4.3.3

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In summary, the conversation is about Section 4.3 of Paul E. Bland's book "Rings and Their Modules" and understanding the proof of part of Proposition 4.3.3. The proof states that if (d) is equal to the sum of several principal ideals, then each element of the sum is also in (d).
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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 help to fully understand the proof of part of Proposition 4.3.3 ... ...

Proposition 4.3.3 reads as follows:View attachment 8247
View attachment 8248
In the above proof by Bland we read the following:

"... ... If \(\displaystyle (d) = a_1 R + a_2 R + \ ... \ ... \ + a_n R\), then each \(\displaystyle a_i\) is in \(\displaystyle (d)\) ... ... "Can someone please explain how \(\displaystyle (d) = a_1 R + a_2 R + \ ... \ ... \ + a_n R\) implies each \(\displaystyle a_i\) is in \(\displaystyle (d)\) ... ..Peter
 
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Hi Peter,
$$a_1 = a_1\cdot 1 + a_2 \cdot 0 + \cdots + a_n \cdot 0 \in (d)$$ and similarly for the other $a_i$.
 

FAQ: Principal Ideal Rings and GCDs .... .... Bland Proposition 4.3.3

1. What is a principal ideal ring?

A principal ideal ring is a type of ring in abstract algebra where every ideal (a subset of the ring closed under addition and multiplication by elements of the ring) can be generated by a single element, called the "generator" or "principal." This is in contrast to other types of rings where ideals may require multiple generators.

2. What is the significance of Bland Proposition 4.3.3?

Bland Proposition 4.3.3 is a theorem that states that in a principal ideal ring, every ideal can be generated by a single element that is a greatest common divisor (GCD) of all the elements in the ideal. This theorem is important because it provides a useful method for finding generators of ideals in principal ideal rings.

3. How is a GCD related to principal ideal rings?

A GCD (greatest common divisor) is related to principal ideal rings because it is used as a generator for ideals in these types of rings. In particular, Bland Proposition 4.3.3 states that every ideal in a principal ideal ring can be generated by a GCD of its elements.

4. Can principal ideal rings have more than one generator?

No, by definition, every ideal in a principal ideal ring can be generated by a single element. Therefore, principal ideal rings can only have one generator for each ideal.

5. How are principal ideal rings different from other types of rings?

Principal ideal rings are unique in that every ideal can be generated by a single element, whereas other types of rings may require multiple generators for ideals. This property makes principal ideal rings useful for solving problems and proofs in abstract algebra.

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