= Local endomorphism ring
An <endomorphism ring> is local when its nonunits form a proper two-sided <ideal>; commutativity is not required. The <Fitting lemma> gives this property for an indecomposable finite-length <module>. If two nonunits had an invertible sum, multiplication by its inverse would give $I=f+g$ with noninvertible $f,g$. Fitting makes $f$ nilpotent, so $I-f=g$ would be invertible, a contradiction. Products with a nonunit are nonunits by finite-length injectivity/surjectivity. The quotient by the nonunit ideal is a <division ring>.
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