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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