Prime Ideal and Proper Ideal

Question

Could someone please explain to me the definition of a prime ideal and a proper ideal. I honestly do not understand this concept.

If possible please explain your version of the definition in the most brain dead way possible. Imagine you are trying to explain this to a little kid, how you would help him understand the concept. I just need a clear and meaningful explanation.

I hope someone will help. Thank you.

Answer 1

Here's an answer in the simplest terms I can manage, using the set (ring) $\mathbb{Z}$ of integers as an example. I'll assume you know what an ideal is.

$3 \mathbb{Z} = \{ \ldots -3, 0, 3, 6, \ldots \}$ is an ideal in $\mathbb{Z}$. It's proper because is isn't all of $\mathbb{Z}$. The only ideal of $\mathbb{Z}$ that isn't proper is $\mathbb{Z}$ itself.

For any integer $n$, the set $n\mathbb{Z}$ of multiples of $n$ is an ideal. It's a proper ideal unless $n = \pm 1$.

The idea of a prime ideal is a little more complicated. The ring of integers is a very nice ring with a very nice property you know about from elementary school: every integer is uniquely the product of prime integers. The proof of that fact depends on a simpler fact: whenever a prime number $p$ divides a product of two numbers it must divide one or the other (or both).

Now let's see what that says about the ideal $p\mathbb{Z}$ when $p$ is a prime number. Suppose $ab \in p\mathbb{Z}$. Then $p$ divides $a$ or $b$ (or both), so $a$ or $b$ (or both) is in $p\mathbb{Z}$.

That's the motiviation for this definition: an ideal $I$ is prime provided that whenever $ab \in I$, one or both of $a$ and $b$ is in $I$.

For the ring of integers, $n\mathbb{Z}$ is a prime ideal just when $n$ is a prime number.

For more general rings, you may not be able to say nice things about prime elements, but can prove theorems about prime ideals. That's why you make that definition.

The wikipedia page https://en.wikipedia.org/wiki/Prime_ideal has examples from other rings.

I don't know whether this explanation would work for a little kid, but I'm pretty sure I could make it make sense for the good sixth graders I hang out with sometimes. I'd have to explain a little first about what a ring was, but they'd like that.

Answer 2

First of all $\mathbf Z_3$ is a ring, not an ideal, much less a prime ideal.

A prime ideal is an ideal $P$ (i.e. a subset of a ring that is stable by addition, and by multiplication by an element of the ring), which further has the property that whenever a product $xy\in P$, at least one of the factors $x,y$ lies in $P$. It is the ideal-theoretic version of the number of a prime number, or an irreducible polynomial.

In terms of quotient rings, an ideal $P$ in the ring $A$ is prime if and only if the quotient ring $A/P$ is an integral domain.

As an example, a natural number $p$ is prime if and only if the principal ideal $p\mathbf Z$ is a prime ideal.

A proper ideal is simply an ideal that is different from the whole ring.

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