Let F be a field and R = Mn(F) the ring of n×n matrices with entires...

Let F be a field and R = Mn(F) the ring of n×n matrices with entires in F. Prove that R has no two sided ideals except (0) and (1).

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Expert Solution

Let be a two sided ideal in . We show that where denotes the identity matrix. Since , there exists a nonzero matrix . Let . Then, by applying some elementary row and column operations on we can transform it to

where is the identity matrix.

Since the elementary row operations are applied just by multiplying by an elementary matrix on left and a column operation by multiplying by an elementary operation on right, we get there exist elementary matrices such that

and this shows that

since is a two sided ideal in .

Now just by multiplying

by permutation matrices on left and then on right we get that contains several matrices having and on diagonal and all other entries and by adding some of them we get that contains a digonal matrix having all diagonal entries nonzero.

Now, if we just multiply that diagonal matrix by an appropriate diagonal matrix, we get that contains the identity matrix and hence it contains al the matrices from which shows that

Colby Messinger answered 1 year ago

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