[GAP Forum] Enumeration in a matrix algebra

[degraaf at science.unitn.it]

Dear Marc, Erik, Forum,

> I have a finitely generated matrix algebra over the rationals that I know is
> not a field but contains
> zero divisors.
> To find one (and this seems to be quite suitable) I simply use:
> i:=First(rc.algebra, a-> Determinant(a)=0 and a <> 0*I);

> Does anybody know how to do this another way with more chance for a success?

In general this is a very difficult problem. If the input basis is
bad, then no general algorithm is known (if the algebra is central simple).
For more information:

@article {MR2325916,
     AUTHOR = {P{\'{\i}}lnikov{\'a}, Jana},
      TITLE = {Trivializing a central simple algebra of degree 4 over the
               rational numbers},
    JOURNAL = {J. Symbolic Comput.},
   FJOURNAL = {Journal of Symbolic Computation},
     VOLUME = {42},
       YEAR = {2007},
     NUMBER = {6},
      PAGES = {579--586},
       ISSN = {0747-7171},
    MRCLASS = {16K20},
   MRNUMBER = {2325916 (2008c:16030)},
MRREVIEWER = {Zinovy Reichstein},
        DOI = {10.1016/j.jsc.2007.01.001},
        URL = {http://dx.doi.org/10.1016/j.jsc.2007.01.001},
}

Best wishes,

Willem de Graaf