Talk:Prime ideal

The previously-listed statement that a subset S of a ring R is a prime ideal iff R\S is closed under multiplication was false. For example, Z\{-1,1} is clearly not an ideal since it's not closed under addition (e.g. 3-2 = 1 is not in it) but {-1,1} certainly is closed under multiplication.

The characterisation of prime ideals at the bottom of the page is incorrect - Dave Benson

I changed the bottom of the page to read that AB is a subset of P instead of AB = P. - Adam Glesser

Cool. Hi Adam. - Dave

Switching definitions?

We should present this as the definition of a prime ideal:

P is a prime ideal iff it is a proper ideal such that for any ideals A, B, $AB\subseteq P\Longrightarrow A\subseteq P\vee B\subseteq P$

This definition works in both the commutative and non-commutative cases, and is equivalent to the definitions we're currently using. It's simpler than the one we're using for the non-commutative case, and as simple as the one we're using for the commutative case; it's also more closely analogous to the standard definition of a prime number, with "proper" taking the place of "not unity" and subset taking the place of divisibility. It should lend the article greater unity and clarity. Any objections? — ciphergoth 13:33, 19 August 2008 (UTC)

Every nonzero ring has a prime ideal?

Surely this is not the case. Consider rings which are fields, which have no (proper) ideals to begin with. —Preceding unsigned comment added by 68.61.156.228 (talk) 06:31, 28 October 2009 (UTC)

The zero ideal is a prime ideal in any field (or integral domain, for that matter). — Emil J. 13:45, 2 November 2009 (UTC)

Zero ideal in ℤ

Thanks for correction. It was my mistake and confusion: {0} is prime, though it is not maximal and is the only ideal giving this combination of properties. Incnis Mrsi (talk) 20:05, 14 January 2013 (UTC)

Wrong caption to image

The caption to the image states that 4Z is a prime ideal. Now 2 * 2 is in 4Z, but 2 is not! Hence 4Z is not primary. 212.127.195.242 (talk) 21:34, 5 October 2013 (UTC)

The caption says that 4Z is not prime, while it is primary. Both claims are correct.—Emil J. 10:08, 7 October 2013 (UTC)

As Emil has already commented, there was nothing factually wrong in the caption, but it certainly was not a very clear caption overall. Beyond that, the graph was not particularly good for this page. (It looks well-suited to the divisibility article, though.) I went ahead and made a new one which I hope is clearer. Rschwieb (talk) 22:53, 7 October 2013 (UTC)

Improving Examples

There should be more explicit examples along with techniques for constructing prime ideals. This should include the use of eisenstein's criterion for analyzing the reducibility of polynomials, constructing complete intersection ideals by taking coprime irreducible polynomials, and giving non-complete intersection prime ideals, such as from the embedding of a projective variety using a twist > n of a very ample line bundle. — Preceding unsigned comment added by 70.59.20.131 (talk) 02:24, 29 August 2017 (UTC)