On regular algebraic systems a note on notes by Iseki, Kovacs, and Lajos
F. M. Sioson · Proceedings of the Japan Academy Series A Mathematical Sciences · 1963
regular rings and semigroups as algebraic systems satisfying the property RL--RL for any right ideal R and any left ideal L. A semigroup (S, .)and a ring or semiring (S, +,-) is regular iff for each s eS there exists an S such that sxs--s.Clearly, this follows from the statement: for each s eS, there exist x, yeS such that sxys --s.The two statements are equivalent, for, if for each s eS there exists an xS such that ss--s, then also there exist a z eS such that x--xzx-x(zx)--xy and therefore sxys-s.