It's multivariate (sorry I didn't specify), so being irreducible and having 
no non-trivial idempotents aren't equivalent.

On Wednesday, April 18, 2012 4:04:04 PM UTC-4, David Loeffler wrote:
>
> On 18 April 2012 15:50, diophan <[email protected]> wrote:
> > I'll check out the reference. If it makes the situation any better the 
> ring
> > is a quotient of a polynomial ring over a finite field.
>
> If it's a quotient of a one-variable polynomial ring, you can just
> factor the defining polynomial, can't you? The quotient will have
> non-trivial idempotents iff the polynomial has more than one
> irreducible factor.
>
> David
>
>

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