[GAP Forum] ring theory

[Alexander Hulpke]

Dear Igor,

> I think you must agree that, even in such trivial cases, there must be no such epic bugs as this one with rings.

I would not consider this as a bug (and would be careful with the `epic' attribute in any case). Of course this is not a ``hard'' problem, but simply one for which the computer is not well suited.

I am not aware of an element test for infinite rings that will always terminate. As I wrote, I can imagine special methods for subrings of polynomial rings or subrings of the integers (or for element test of the generators), but at this point I see minimal gain in implementing this and am rather doubtful of the merits of devoting time on such particular features. Thus again my request for any concrete need of such an element test (the use itself can be trivial, but the context shouldn't be) for rings.

Best,

  Alexander Hulpke

> 22.01.2011, 19:41, "Alexander Hulpke" <ahulpke at gmail.com>:
>> Dear Forum,
>> 
>> On Jan 21, 2011, at 10:23 PM, Igor Korepanov wrote:
>> 
>>>  gap> r := Ring( 10, 12 );
>>>  <ring with 2 generators>
>>>  gap> 10 in r;
>>> 
>>>  - and GAP is still thinking over this very profound question.
>>>  What did I miss?
>> 
>> The element test implemented fro rings in GAP first determines the element list of the ring and thus does not terminate for infinite rings. While I one could combine the element test in the computation of the element list and terminate once the element has been found, this still would be unsatisfactory in any example beyond trivial ones and not terminate if the element was not in the ring.
>> 
>> If there is a genuine need for element tests in infinite rings (beyond toy examples) I would be interested to know for what cases, since for situations such as polynomial rings alternatives might be possible.
>> 
>> Best,
>> 
>>    Alexander Hulpke