Simple Ring: Artenian Ring
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A simple ring is an Artenian ring with no 2-sided ideals. Let K be a field. Let V be an n-dimensional vector space over K and let A = EndK(V), the ring of n x n matrices over K. The ring is clearly Artenian since it has finite dimension over K. We want to see that the 2-sided ideal are either (0) or A. Let W be a subspace of V. Then LW = {f 2 A | f (W) = 0} is clearly a left ideal of A. Let I be a left ideal, W = { x2 V | f (x) = 0 for all f 2 I}, so I LW. Let e1, . . . , en be a basis
of V with e1, . . . , er a basis of W.
Show that if f(x) has zero e1 - coordinate for all f 2 I, then x2 W. Note that if permutes the ei, then f 2 I.
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This solution helps with a problem involving an Artenian ring. Step-by-step calculations are provided.
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First note that we can identify the elements of the ring A with the linear transformations VV. Moreover, such linear transformations are uniquely determined by their images on e_1, e_2,...,e_n.
Let x∈V and f(x) has a zero ...
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