Some remarks on topologically semiprime ideals

Let 5 be a topological semigroup.(An algebraic semigroup is a topological semi group with discrete topology].An ideal i of .S is called topologically semiprime if whenever x ф1, then Г(х) n / = 0, where Г(х) is the compact monothetic semigroup generated by x e 5, that is, Г{х) = {x"}^=:i.An ideal Q oi S is called a group-ideal if whenever x ф Q, then J(x), the principal ideal generated by x, contains a group G which misses Q.In this note, we shall study the class of topologically semiprime ideals and their relationship with group-ideals.It is shown that topologically semiprime ideals are topological generalizations of semiprime ideals and completely semiprime ideals.In particular, an ideal of a compact semigroup is topologically semiprime if and only if it is an intersection of completely open prime ideals.This result will bring together and generalize many results in the literature concerning prime ideals and their intersections.For the definitions of prime ideals, semiprime ideals, completely prime ideals, completely semiprime ideals and other terminology used in this paper, the reader is referred to A. H. Clifford and G. B. Preston [1].Unless otherwise stated, the word "ideal" means two sided ideal of S.The following theorem characterizes the group-ideals in a compact semigroup.