Small skew lattices in rings (Q1780017)

In the theory of semigroups (rings) the role of the set \(E\) of idempotents is known, especially when any two idempotents \(e\) and \(f\) commute (i.e. \(ef= fe\)) or are central (i.e. \(xe= ex\) for every \(e\in E\) and arbitrary \(x\)). The author shows that for given idempotents \(e\) and \(f\) in a ring \(R\), if \(ef\) and \(fe\) are also idempotents, then \(e\) and \(f\) generate a skew lattice \(\langle e,f\rangle\) under the operations \(\vee\) and \(\wedge\) (i.e. both \(\vee\) and \(\wedge\) are associative and satisfy the absorption identities). In addition, \(\langle e,f\rangle\) is a direct product \(U\times A\) of a lattice \(U\) and a rectangular skew lattice \(A\) (i.e. \(x\vee y= y\wedge x\)) and each order is dividing 4.