maximal ideals

[viola crescentini]

Hi!

I would like to know if there is a command that checks whether an ideal M is maximal. And when an ideal is M-primary.

Sorry for the inconvenience.

[viola crescentini]

I try to check if R/M is a field with isField, but for example:

• R=QQ[x]
• I=ideal(x^2+1)
• A=R/I
• isField A
• false

Why?

[Michael Stillman]

isField is not really doing what you want.  Take a look at this doc:

  viewHelp isField

(although the doc isn’t perfect) says that the routine doesn’t do computation to determine if it is a field.

(In fact, this doc has your exact example!).

The purpose of ‘isField’ is more as an internal function: if a ring is a field, then the Groebner basis functions will handle it differently.

This function indicates whether it can do so.  In order to use your ring A in this way:

kk = toField A — in your example below

isField kk — will return true.

'isField’ is just returning whether M2 things of this as a field.

I agree that this is misleading.  Perhaps we should change it, so that there is an internal function, called something like:

isDeclaredField — current functionality

and then change isField:

isField — actually computes whether the ideal I (in your example) is a maximal ideal.

Changing this will break current code that depends on ‘isField', but perhaps almost no one is using this function at the moment (except in internal M2 code).

— mike

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[viola crescentini]

Oh ok. So currently there is no way to check if an ideal is maximal.

Thanks anyway.

[Michael Stillman]

True, but  if A is a polynomial ring, then I is maximal if (a) I is prime, and (b) codim I == numgens A.

Here is a function in M2 to do this in a polynomial ring (or quotient of one).  I’ll plan on putting this

into M2, but first need to make sure it works with all rings (or gives errors for rings it can’t handle).

isMaximal = method()

isMaximal Ideal := (I) -> (

    (J,F) := flattenRing I;

    isPrime J and codim J === numgens ring J

    )

On Feb 12, 2014, at 10:43 AM, viola crescentini <cresce...@gmail.com> wrote:

Oh ok. So currently there is no way to check if an ideal is maximal.

Thanks anyway.

[viola crescentini]

Unfortunately I need both cases: the case of a polynomial ring and a ring.

[Baptiste Calmès]

Dear Mike,

Has the method "isMaximal" that you gave above made it to M2 yet, maybe under another name or avatar?

As the OP, I'm interested in efficient ways to test whether a quotient of a polynomial ring is a field.

In my specific example, the method you suggest above fails because isPrime fails because it calls minimalPrimes that has no strategy to work. I think it's due to the fact that the coefficient ring of my polynomial ring is not simple enough.(it is a field of fractions in two variables over a finite field).

Since my ideal is generated by one element, I can nevertheless test whether it is prime by applying isPrime directly to the polynomial generating it, which gives a result.

But, I guess on top of the question above, I'm interested in knowing how much effort has been put into making sure the method "isPrime(Ideal)" works in a wide variety of situations. It is a very important core method in M2, isn't it?

What is the status of its development?

All the best,

Baptiste