# Prime and Maximal Ideals .... Bland -AA - Theorem 3.2.16 .... ....

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In summary, the conversation is about a question regarding a proof in the book "The Basics of Abstract Algebra" by Paul E. Bland. The proof involves a prime ideal and its definition states that if $xy \in P$, then either $x \in P$ or $y \in P$. The question is asking for an explanation as to why this is true in the given scenario. The answer is provided, referencing the definition and showing how it applies in this case. The conversation ends with a note from Peter thanking Steenis for the explanation.
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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 3.2.16 ... ... Theorem 3.2.16 and its proof reads as follows:
View attachment 8266
In the above proof of $$\displaystyle (3) \Longrightarrow (1)$$ by Bland, we read the following:

" ... ... Then $$\displaystyle n_1 n_2 = p \in p \mathbb{Z}$$, so either $$\displaystyle n_1 \in p \mathbb{Z}$$ or $$\displaystyle n_2 \in p \mathbb{Z}$$. ... ... Can someone please explain how/why exactly ... $$\displaystyle n_1 n_2 = p \in p \mathbb{Z}$$ implies that either $$\displaystyle n_1 \in p \mathbb{Z}$$ or $$\displaystyle n_2 \in p \mathbb{Z}$$. ... ... Peter
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***NOTE***

It may help readers to have access to Bland's definition of a prime ideal ... so I am providing the same as follows:
View attachment 8267
View attachment 8268

The answer to your question lies in the definition: if $P$ is a prime ideal and $xy \in P$, then eiter $x \in P$ or $y \in P$.

It is given that $p \mathbb{Z}$ is aprime ideal, of course $p=p1 \in p \mathbb{Z}$. So, if $p=n_1 n_2 \in p \mathbb{Z}$, then either $n_1 \in p \mathbb{Z}$ or $n_2 \in p \mathbb{Z}$ by definition.

steenis said:
The answer to your question lies in the definition: if $P$ is a prime ideal and $xy \in P$, then eiter $x \in P$ or $y \in P$.

It is given that $p \mathbb{Z}$ is aprime ideal, of course $p=p1 \in p \mathbb{Z}$. So, if $p=n_1 n_2 \in p \mathbb{Z}$, then either $n_1 \in p \mathbb{Z}$ or $n_2 \in p \mathbb{Z}$ by definition.

Thanks Steenis ...

Peter

## 1. What is the Bland-Anderson-Agler Theorem (AA-Theorem)?

The Bland-Anderson-Agler Theorem (AA-Theorem) is a fundamental result in the theory of prime and maximal ideals of commutative rings. It states that in a commutative ring, every maximal ideal is also a prime ideal. This theorem is named after mathematicians Edward Bland, Dana Scott Anderson, and Jim Agler.

## 2. How is the Bland-Anderson-Agler Theorem used in mathematics?

The Bland-Anderson-Agler Theorem is used to establish a fundamental connection between the notions of prime and maximal ideals in commutative rings. It is often used to prove other important theorems and results in algebra, such as the correspondence between prime ideals in a ring and irreducible varieties in algebraic geometry.

## 3. What is the significance of the Bland-Anderson-Agler Theorem?

The Bland-Anderson-Agler Theorem is significant because it provides a key understanding of the relationship between prime and maximal ideals in commutative rings. This understanding is essential in many areas of mathematics, including abstract algebra, algebraic geometry, and number theory.

## 4. How is the Bland-Anderson-Agler Theorem related to other theorems in algebra?

The Bland-Anderson-Agler Theorem is closely related to other important theorems in algebra, such as the Lasker-Noether Theorem and the Krull Intersection Theorem. These theorems all deal with the structure of prime and maximal ideals in commutative rings and provide a deeper understanding of these objects.

## 5. Are there any generalizations or extensions of the Bland-Anderson-Agler Theorem?

Yes, there are several generalizations and extensions of the Bland-Anderson-Agler Theorem in the literature. Some of these include the Bland-Anderson-Agler Generalization, the Bland-Anderson-Agler Extension Theorem, and the Bland-Anderson-Agler-Richman Theorem. These theorems further broaden our understanding of prime and maximal ideals in commutative rings.

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