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    Let K be a field. (All of this works just as well for a division ring.) Let V be an n-dimensional vector space over K and let A = EndK(V), the ring of n n matrices over K. The ring A is clearly Artinian since it has finite dimension over K. We need to see that the 2-sided ideals are either (0) or A.

    1. Let W be a subspace of V. Then LW = { f  A | f (W) = 0 } is clearly a left ideal of A.
    Prove that every left ideal has this form.

    Here is an outline of one method. Let I be a left ideal, W = { x  V | f (x) = 0 f  I },
    so I  LW. Let e1, . . . , en be a basis of V with e1, . . . , er a basis of W.

    a. Show that if f (x) has zero e1-coordinate for all f  I, then x  W. [Note that if π permutes the ei, then π ◦ f  I.]

    b. Note that H = {e1-coordinate function of f | f  I } is a subspace of V * = hom(V, K).
    Use duality and (a) to conclude that H is the annihilator of W, and so any linear function
    V → K vanishing on W is in H.

    c. Let Eij be the function taking ej to ei and vanishing on ek for k ≠ j. Using (b) and a projection onto e1, prove that E1j  I for j > r.

    d. Permute the ei to show all Eij  I for j > r.

    e. Show these elements span LW.

    2. Prove that (LW)T = LT - 1W (L sub T-1W) for any T  A. Deduce that (0) and A are the only 2-sided ideals of A, so that matrix rings are simple.

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    Solution Preview

    1. Let be a subspace of . Then is clearly a left ideal of . Prove that every left ideal has this form.
    Suppose is a left ideal of , . Then for any , we have . So . Let be a basis of with a basis of .
    a. I show that if has zero -coordinate for all , then .
    Consider any , since is a left ideal, then for any permutation matrix , we have .
    Now has zero -coordinate for all , I claim that . If not, then we can find some , such that . Then must have some non-zero coordinate, say ...

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