Remark. three levels of splitting [fgap-0012]
Remark. three levels of splitting [fgap-0012]
The same formulas support three conclusions, but their structures should not be conflated: \[ \begin {array}{c|c|c} \text {level}&\text {conclusion}&\text {input}\\ \hline \text {additive}&M=e_+M\oplus e_-M& e_++e_-=1,\ e_+e_-=e_-e_+=0\\ \text {module}&M_\pm \text { are eigensubmodules}& \text {central action of }z\\ \text {algebra}&A\cong e_+A\times e_-A& e_+,e_-\text { central}. \end {array} \]
For an arbitrary idempotent \(e\in A\), left multiplication by \(e\) and \(1-e\) gives a split of the underlying \(R\)-module, \[ A=eA\oplus (1-e)A, \] and therefore also of the underlying additive group. Without centrality, these projections need not be homomorphisms of the left regular \(A\)-module: \(eA\) and \((1-e)A\) are right ideals, but need not be left ideals or two-sided ideals. The two summands therefore need not be algebra factors. The product theorem is stronger than this \(R\)-module direct-sum statement.