While in general it is not true that an ideal of an ideal is an ideal, this proof in Humphreys confuses me. Could someone please explain to me why the underlined sentences are true?
2026-05-06 03:17:38.1778037458
Reading help, ideal of an ideal is an ideal?
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take an element $i$ in an ideal $I$ of $L_1.$ Notice that for any $l\in L_1, m \in L_1^\perp,$ $[l, m]=0$ (this is true for any two ideals with trivial intersection). Now, writing $l\in L = l_1 + m,$ we see that $[i, l] = [i, l_1] + [i, m] = [i, l_1] \in I.$