localizations of Zmodn

Stream: new members

[Madison Crim]

Is there anything in mathlib about localizations of Zmodn?

[Kevin Buzzard]

Are all localisations quotients for n>0? If so, what's the general class of rings for which this is true?

[Damiano Testa]

It looks like products of rings whose radicals are fields satisfy the condition, right?

[Kevin Buzzard]

Yes this sounds right. So @Madison Crim you should prove that any ring with this property (I guess the products have to be finite) has all localisations being projections and then prove that ZMod n has this property

[Madison Crim]

@Damiano Testa @Kevin Buzzard I've given this some thought and I'm confused on how the radical could be a field. I assume by this you mean Jacobson radical. However, the Jacobson radical can't contain the identity, so I don't see how it could be a field.

[Kevin Buzzard]

I guess Damiano meant (finite) products of rings which have the property that if you quotient by the nilradical you get a field. Examples: $\Z/p^n\Z$ and products (so $\Z/N\Z$ by CRT).

[Madison Crim]

Okay, so in particular then the product of local rings since this would imply the nilradical is a maximal ideal.

[Damiano Testa]

Yes, Kevin's interpretation is correct: I meant products of rings whose nilradicals are maximal (i.e. such that the quotient by the nilradical is a field).

[Kevin Buzzard]

Re local rings: No, for a local ring like $\R[[x]]$ the nilradical is zero. It's zero-dimensional local rings or something like that.