- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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

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

***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

Sorry about the legibility ... but Bland shades his definitions ...Peter