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Unformatted text preview: L can be expressed uniquely as a product of simple ideals.
Proof. First we should point out that if L is a direct sum of two ideals L = J1 ⊕ J2 then
L ∼ J1 × J2 with the isomorphism given by the projection maps L → L/J2 , L → L/J1 .
If L is simple, the statement trivially holds. Otherwise, let J ⊆ L be a minimal ideal.
Deﬁne J ⊥ to be the set of all x ∈ L so that κ(x, J ) = 0. Then J ⊥ is an ideal since
κ([xy ], J ) = κ(x, [yJ ]) ⊆ κ(x, J ) = 0 for all x ∈ J ⊥ , y ∈ L. Therefore, J ∩ J ⊥ is also an
ideal. By minimality of J we have either J ∩ J ⊥ = J or J ∩ J ⊥ = 0. The ﬁrst case is
not possible si...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11