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The injective dimension of a noetherian local ring is either infinite or equal to its dimension
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Since dim B=0, it's artinian and we can talk about its minimal ideals. Such an ideal is principal of the form but (0:b) is a maximal ideal because Bb has no submodule (B is a quotient of A and (0:b) is a quotient of m) therefore it's isomorphic to k.
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B is Artinian, so that 
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B is Artinian, so by a theorem in Sharp's book, a power of the maximal ideal annihilates the ring; hence the maximal ideal is a subset of zero devisors so belongs to Ass, therefore .