R is not in Ideals(R)

[tscrim]

This was surprising to me:

sage: ZZ in Ideals(ZZ)
False

Should be easy enough to fix as it is just a missed case in the custom __contains__ method.

Component: categories

Issue created by migration from https://trac.sagemath.org/ticket/22722

[0adfa02f-507d-46c6-aa3b-f2437a2a5873]

comment:1

This is consistent with many other functions that do not treat a ring R as an ideal, for example

sage: ZZ.quotient_ring(ZZ)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
...
TypeError: I must be an ideal of ZZ or an integer
sage: ZZ.quotient_ring(1*ZZ)
Ring of integers modulo 1

So ZZ is not treated as an ideal, but 1*ZZ is. And this is correctly implemented:

sage: (1*ZZ) in Ideals(ZZ)
True

[tscrim]

comment:2

However, that doesn't absolve the problem. I would say it is evidence that it is more systematic. I think I will start digging into this on the code side.

I guess another question is should 1*ZZ == ZZ and/or be ZZ?

[0adfa02f-507d-46c6-aa3b-f2437a2a5873]

comment:3

It is indeed systematic and this sort of thing afflicts other categories, like monoids

sage: ZZ in Monoids()
True
sage: 1*ZZ in Monoids()
False

semigroups,

sage: ZZ in Semigroups()
True
sage: 2*ZZ in Semigroups()
False

and rings:

sage: (1*ZZ) in Rings()
False
sage: (1*ZZ) in Rngs()
False

On the plus side, Sage knows that ideals are additive groups:

ZZ.categories()[24]
Category of additive groups
sage: AG = _
sage: AG.axioms()
frozenset({'AdditiveAssociative', 'AdditiveInverse', 'AdditiveUnital'})
sage: ZZ in AG
True
sage: (2*ZZ) in AG
True

To weigh in on your question, if you make ZZ known as an ideal of itself, then 1*ZZ == ZZ would make sense, and indeed they could be identical.