Talk:Glossary of commutative algebra

Separable algebra

The definition here is "An algebra over a field is called separable if its extension by any finite purely inseparable extension is reduced". The definition at Separable algebra is "associative K-algebra A is said to be separable if for every field extension $\scriptstyle L/K$ the algebra $\scriptstyle A\otimes _{K}L$ is semisimple". The latter is supported by, for example, DeMeyer and Ingraham (1971). Are these the same and if so what is the reference? Spectral sequence (talk) 17:36, 10 August 2013 (UTC)

See a related sentence in "Perfect closure and perfection" at perfect field. My "guess" is they are likely equivalent, but I'm not sure. -- Taku (talk) 19:31, 10 August 2013 (UTC)

Cohen–Macaulay

Sorry, this really doesn't make sense:

A ring is called Cohen–Macaulay if all local rings are Cohen–Macaulay, meaning that they are Noetherian local rings with dimension equal to their depth.

I think it is intending to say something like

A local ring is Cohen–Macaulay if it is Noetherian with dimension equal to depth. In general a ring is called Cohen-Macaulay if the localisation at every maximal ideal is Cohen-Macaulay.

This is the definition of Burns and Herzog (1998) p.57. Spectral sequence (talk) 11:36, 11 August 2013 (UTC)

"Proper", as in "proper ideal"

needs defining. Johnbod (talk) 21:27, 6 December 2013 (UTC)

No, it just means proper as in a proper subset. (unsigned))

which is also NOT DEFINED, and I suspect incorrect - proper ideal is more relevant and I think different. Whichever way you look at it, more is needed here. I'm trying to find the best link for the mathmatical sense of proper right, but WP seems to be no help at all. Johnbod (talk) 03:19, 7 December 2013 (UTC)

What makes you think "proper" in "proper ideal" is something tricky? (see proper ideal for a definition.) It's not defined here, because it has only the expected meaning; i.e., proper as in "proper subset". -- Taku (talk) 14:25, 7 December 2013 (UTC)

Your idea of "definition" is clearly very different from mine (or the dictionaries'). Johnbod (talk) 15:22, 7 December 2013 (UTC)