Let R be a ring with identity and S the ring of all n×n matrices over R,J is an ideal of S if and only if J is the ring

[answerhappygod]

Let R be a ring with identity and S the ring
of all n×n matrices over R,J is
an ideal of S if and only if J is the ring of all
n×n matrices over I for some ideal
I in R. [Hint: Given J, let I
be the set of all those elements of R that appear as the
row 1 -column 1 entry of some matrix in J. Use the
matrices Er,t, where
1≤r≤n,1≤s≤n, and
Er,s has 1R as the row
r-column
s entry and 0 elsewhere.
Observe that for a matrix
A=aij,Ep,rAEs,4
is the matrix with are in the row
p-column q entry and 0 elsewhere.]