Rings- units, nilpotents, idempotents

[missavvy]

Homework Statement

Find the units, nilpotents and idempotents for the ring R =
[\Re \Re]
[0 \Re]

(Those fancy R's are suppose to be the set of Reals by the way.. not good with this typing math stuff)

Homework Equations

The Attempt at a Solution

I'm not actually sure I understand the ring itself. So it is a matrix with entries \Re in the 1-1, 1-2, 2-2 positions and 0 in the 2-1 position..
So is it the entire set of the real numbers?? :S

Anyways for the units, I said all elements are units except for when \Re=0.

But then again I wasn't sure if I'm suppose to use the entire set as those positions, or is it just random numbers from the reals? like a,b,c. OR is it just any number from the reals, but each position has the same #, say x belonging to the reals?

Eughh.. just clarification on the actual question I guess is what I need some help with.

Thanks :)

 

[Dick]

It's got to be just all matrices [[a,b],[0,c]] where a, b, and c are real numbers. It's your 'random numbers from the reals' theory. Where can you go from there?

 

[missavvy]

ok well for the units I'm attempting to solve some systems of equations..
[[a,b],[0,c]]*[[a',b'],[d',c']] = [[1,0],[0,1]]
and
[[a',b'],[d',c']]*[[a,b],[0,c]] = [[1,0],[0,1]]

but for matrices, isn't everything invertible whose det is not 0?

 

[Dick]

ok well for the units I'm attempting to solve some systems of equations..
[[a,b],[0,c]]*[[a',b'],[d',c']] = [[1,0],[0,1]]
and
[[a',b'],[d',c']]*[[a,b],[0,c]] = [[1,0],[0,1]]

but for matrices, isn't everything invertible whose det is not 0?

Sure, if det is nonzero then the matrix is invertible. All you have to show is that the inverse is also in the ring.