 Every right and left ideal is a direct summand of "R", and so is of the form "eR" or "Re" where "e" is an idempotent of "R".Paralleling this example, von Neumann regular rings are seen to be both right and left Bézout rings.If "D" is a division ring and formula_24 is a ring endomorphism which is not an automorphism, then the skew polynomial ring formula_25 is known to be a principal left ideal domain which is not right Noetherian, and hence it cannot be a principal right ideal ring.