Integer ideals

Algebra Level 4

A nonempty set $I$ in a ring $R$ is called an ideal if

(1) it is closed under addition: $a\in I, b\in I \Rightarrow a+b \in I$

(2) it "swallows up" under multiplication: $a \in R, i \in I \Rightarrow ai \in I$ .

A proper ideal is one that is not equal to the entire ring.

A proper ideal is prime if $ab \in I$ implies that $a\in I$ or $b \in I$ .

A proper ideal is maximal if there are no ideals in between it and the entire ring: if $J$ is an ideal, then $I \subseteq J \subseteq R$ implies $I=J$ or $J=R$ .

How many ideals of $\mathbb Z$ are prime but not maximal?

It should help to read the ring theory wiki!

1 Infinitely many 2 0

1 solution

展豪張
Mar 20, 2016

Only $(0)$ is prime but not maximal.

Agreed! Proof? :)

Patrick Corn - 5 years, 2 months ago

First of all, $\mathbb Z$ is a PIR .
Consider special cases $(0)$ and $(1)$ first. $(0)$ satisfies requirement while $(1)$ is maximal.
All other cases can be written as $(a)$ where $a$ is either prime or composite.
For prime $a$ , $(a)$ is maximal.
For composite $a$ , $(a)$ is not prime.
Hence only $(0)$ is prime but not maximal.
:)

展豪張 - 5 years, 2 months ago

oops. ok (0) is a prime ideal. i missed this point.

Srikanth Tupurani - 2 years, 6 months ago

Integer ideals

It should help to read the ring theory wiki!

1 solution

0 pending reports