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I have this ring of matrixes: R = { \[ \begin{pmatrix} a & 0\\ b & c\\ \end{pmatrix} \]}

while a,b,c is from some field F.

now, I need to find all the ideals of this ring. I found five ideals. here there are:

i1 = { \[ \begin{pmatrix} 0 & 0\\ b & 0\\ \end{pmatrix} \]}

i2 = { \[ \begin{pmatrix} a & 0\\ b & 0\\ \end{pmatrix} \]}

i3 = { \[ \begin{pmatrix} 0 & 0\\ b & c\\ \end{pmatrix} \]}

i4 = R

i5 = {0}

now, Im kind of stuck to explain why there cannot be a six's ideal. I know intuitively why there cannot be another ideal but its like I can't figure out how formally explain it. I feel like I am dancing around the answer for hours but can't make it right on the spot.

any help?

thank you!