= Maximal left ideal
{title2=$L\subsetneq A$}
A <maximal left ideal> is a proper <left ideal> contained in no larger proper <left ideal>. In a unital <Banach algebra> it is <norm> <closed>: its closure is a <left ideal>, and if that closure were the entire algebra it would contain elements of $L$ within distance less than one of $1$, hence invertible by the <Neumann series>. A proper <left ideal> contains no invertible element. The quotient $A/L$ is consequently a <quotient Banach space> and a <simple module>.
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