What are all the twosided ideals of the two ring R MatzZ2 o

What are all the two-sided ideals of the two ring R = Mat_z(Z_2) of 2 Times 2 matrices over the two-element field, Z_2?

Solution

Matrices of 2x2 have only 0 and R as ideal two sided.

No other two sided ideals are possible.

If possible let C be an ideal which contains a matrix with non zero entry a_ij.Multiply this with identity matrix we get

except i th row all zeros.

Similarly multiply by suitable 2x2 with element 1, 0,0, 1 arranged suitably to make multiplication as only jth column non zero.

Now have a matrix from C which contains only a_ij as non zero entry.
When we try to show that C must contain all M_2x2 matrices this shows that only 0 and entire ring are the two ideals.