Re: A strange ring, and some strange ideals...

Robert Israel wrote:

In article <44581633$0$29227$ba620e4c@xxxxxxxxxxxxxx>,
Marc Bogaerts <mbg.DELSPAMnimda@xxxxxxxxx> wrote:
Let a, b, c be rationals.

Let R be the set of matrices

[ [a, c, b],
[b, a, c],
[c, b, a] ]

For brevity, let me write this as T(a,b,c).

R is clearly a commutative ring.

Not quite obvious to me, but yes it works.

now consider the set I of elements of R of the form

[ [a, a, a],
[a, a, a],
[a, a, a] ]

I is clearly an ideal of R. Is it a maximal ideal?

Hint: The question is whether there is any 2-dimensional ideal
containing I. If the ideal is spanned by I and some element w,
write out the equations for w T(0,1,0) to be in the ideal, and
solve. The fact that z^2 + z + 1 has no rational roots will come
in handy...

Now consider the set J of elements of R which satisfy

a+b+c = 0

J is clearly an ideal of R. Is is a maximal ideal?

Hint: it's maximal not just as an ideal but as a proper
subspace.

You mean as a |Q module?

Are I and J coprime?

That's an easy one.

Indeed, I + J = R.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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