On radicals in semigroups
František Kmeť · Czech digital mathematics library · 1982
FRANTlSEK KMEfLet S be a semigroup and JcSa two-sided ideal of S. All ideals in the following are supposed to be two-sided.An element x e S is called nilpotent with respect to / if x n e J for some positive integer n.An ideal, or a subsemigroup J of S, is called nilpotent with respect to /if FcZ for some positive integer n.An ideal J of S, each element of which is nilpotent with respect to /, is called a nilideal with respect to /.An ideal I, each finitely generated subsemigroup of which is nilpotent with respect to /, is called a locally nilpotent ideal with respect to /.An ideal P of S is called prime if for any two ideals A, B of S AB c P implies that either AcPorBcP.An ideal P of S is called completely prime if for any two elements a, beSabeP implies that either a e P or b e P. A subset Mc S is called a filter of S if P= S -M is a completely prime ideal of S or M= S. It is known that a subset Mof S is a filter of S if and only if x, y e M is equivalent to xyeM.The set of all nilpotent elements of S with respect to an ideal / of S will be denoted by Nj(S). The unionRj(S) of all nilpotent ideals of S with respect to / is called the Schwarz radical of S with respect to /.The union Lj(S) of all locally nilpotent ideals of S with respect to / is called the Sevrin radical of S with respect to /.The union R%S) of all nilideal of S with respect to / is called the Clifford radical of S with respect to /.The intersection Mj(S) of all prime ideals of S which contain / is called the McCoy radical of S with respect to /.The intersection G(S) of all completely prime ideals of S which contain / is called the Luh radical of S with respect to /.J. Luh [5, Theorem 3,3; 3,4; 3,5; 3,7 and Corollary] proved for a semigroup S with the kernel K (the minimal ideal of S) that R K (S) c M K (S) c R%S) c C K (S) and for a commutative semigroup R K (S) = M K (S) = R%(S) = C K (S).§. Schwarz [6, section II, Theorems 7, 8, 9] studied questions connected with the existence of the kernel of a semigroup S and the relations R K (S) c M K (S) c R%(S) c M* (the last relation if S 2 = S and M* ± 0), where M* is the intersection of all maximal ideals of S. R. Sulka [7, Lemma 19] and J. Bosak [2, Theorem 2] proved that in an arbitrary semigroup S with an ideal /cSwe have