Semigroup Ideals

In the literature of ideals a left ideal L of a system S has usually been defined by the inclusion of SL ⊆ L and a right ideal R by the inclusion RS ⊆ R. On the other hand a left zero element z is usually defined by the equation za = z and a right zero element by the equation az = z ; similarly a left or right identity element e is defined by the equation ea = a or ae = a respectively. The author has in this paper made the terms left and right when applied to ideals consistent with the use of the terms left and right when applied to zeros or identities; thus L is a left ideal of a multiplicative system S if LS ⊆ L ; and R is a right ideal of S if SR ⊆ R. The reader is therefore cautioned to interchange the words left and right throughout this paper when referring to other literature on the subject. If such an interchange of words is effected, no change in the symbolism is required.