Principal Ideals and Prime and Maximal Ideals .... Blamnd, AA, Theorem 3.2.19 .... ....

In summary: This shows that $yR = xR$. In summary, the proof shows that if $y \in xR$ and $yR$ is an ideal such that $xR \subseteq yR \subseteq R$, then $xR = yR$.
  • #1
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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 Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows:
View attachment 8269In the above proof by Bland we read the following:"... ... If \(\displaystyle y \in xR\) it immediately follows that \(\displaystyle xR = yR\) ... ... Can someone please explain exactly why \(\displaystyle y \in xR\) implies that \(\displaystyle xR = yR\) ... ... Peter
 
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  • #2
Peter said:
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 Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows:
In the above proof by Bland we read the following:"... ... If \(\displaystyle y \in xR\) it immediately follows that \(\displaystyle xR = yR\) ... ... Can someone please explain exactly why \(\displaystyle y \in xR\) implies that \(\displaystyle xR = yR\) ... ... Peter

I've been thinking about my question ... and think i have an answer ... as follows ...... to show \(\displaystyle y \in xR \Longrightarrow xR = yR\) ...Assume \(\displaystyle y \in xR\) ...

But then we also have \(\displaystyle y \in yR\) ...

\(\displaystyle \Longrightarrow xR \subseteq yR\) ... ... ... ... (1)Now ... let \(\displaystyle a \in yR\) ... in addition to our assumption that \(\displaystyle y \in xR\) ...

then \(\displaystyle a = yb\) for some \(\displaystyle b \in R\) ... ... ... ... (2)

But \(\displaystyle y \in xR\) so \(\displaystyle y = xc\) for some \(\displaystyle c \in R\) ... ... ... ... (3)

Now ... (2) (3) \(\displaystyle \Longrightarrow a = xcb = xd\) where \(\displaystyle d \in R\)

So ... \(\displaystyle a = xd \in xR\) ...

\(\displaystyle \Longrightarrow yR \subseteq xR\) ... ... ... ... (4)Therefore (1) (4) \(\displaystyle \Longrightarrow xR = yR\) ...
Is the above proof correct?Peter
 
  • #3
(4) is correct, but I am afraid that (1) is not correct.

In the proof it is supposed that $yR$ is an ideal such that $xR \subseteq yR \subseteq R$.

Furthermore, $xR$ is a nonzero ideal and $y \in xR$.

$xR$ is an ideal, so for for all $r \in R$ we have $yr \in xR$, thus $yR \subseteq xR$.
 

1. What are principal ideals?

Principal ideals are ideals in a commutative ring that are generated by a single element. They are written as (a) where a is the generator of the ideal.

2. How are principal ideals different from prime ideals?

While principal ideals are generated by a single element, prime ideals are not. A prime ideal is an ideal in a commutative ring where the product of any two elements in the ideal is also in the ideal. This means that prime ideals are closed under multiplication, while principal ideals are not necessarily.

3. What is the relationship between prime and maximal ideals?

Prime ideals are always contained in maximal ideals. A maximal ideal is an ideal in a commutative ring that is not properly contained in any other ideal. In other words, there are no larger ideals that contain the maximal ideal. This means that maximal ideals are the largest possible ideals in a commutative ring.

4. What is the significance of Theorem 3.2.19 in the context of principal and prime and maximal ideals?

Theorem 3.2.19 states that in a commutative ring, every prime ideal is a maximal ideal if and only if the ring is a PID (principal ideal domain). This means that in a PID, every prime ideal is generated by a single element, making it a principal ideal. This is a useful property in the study of commutative rings.

5. Can a commutative ring have multiple maximal ideals?

Yes, a commutative ring can have multiple maximal ideals. In fact, in certain cases, such as in the ring of integers, there are infinitely many maximal ideals. This is because maximal ideals correspond to prime numbers in the ring of integers, and there are infinitely many prime numbers.

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