Talk:Prime ideal

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Mathematics rating:	Start Class	High Priority	Field: Algebra
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Needs references. Geometry guy 01:19, 23 June 2007 (UTC)

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

[edit] 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,

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)