Answer to Question #17269 in Algebra for sanches

Question #17269
Show that if I in R is a modular left ideal, then I can be embedded in a modular maximal left ideal of R
Expert's answer
Fix the element e above. Forany left ideal I' ⊇ I, we have I'is not equal R iff e is not in I'. (For the “only if”part, note that if e ∈ I', then,for any r ∈ R, we have r ∈ re + I ⊆ I'.) Therefore, we can apply Zorn’s Lemma to thefamily of left ideals {I' : I is not equal R and I_⊇ I} to prove the existence of a maximal left idealm containing I. Clearly, m is also modular, and e is not in m.

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