Radicals and their left ideal analogues in a semigroup
František Kmeť · Czech digital mathematics library · 1988
The first section of the present paper deals with an i?*NC-semigroup.It is known that an i?*1VC-semigroup is a semilattice of archimedean semigroups (see [5]).We prove that the converse is also true (Theorem 1).In the second section we prove that in a semigroup S for any left ideal L we have L c r (L) c m (L) _= r*(L) c N(L) c C (L) (Theorem 2).This is a left-sided analogue of the known result about radicals of R. §ulka [9, Lemma 19] and J. Bosak [2].We give some definitions (the others can be found in [2), [3], [7], or [9]).Let S be a semigroup.A non-empty subset / of S is a two-sided (or left) ideal if S ] JS ] ^ J (or S ] J ^ J).The principal two-sided (or left) ideal of S generated by an element aeS is denoted by J(a) (or L(a)).An element xeS is nilpotent with respect to a subset A if x n eA for some positive integer n.The set of all nilpotent elements of S with respect to A is denoted by N(A).A two-sided (or left) ideal A is a nilideal (or left nilideal) with respect to a two-sided (or left) ideal / if A _= N(J).The union of all two-sided (or left) nilideals with respect to a two-sided (or left) ideal J is denoted by R*(J) (or r*(J)).A two-sided (or left) ideal A is nilpotent with respect to a two-sided (or left) ideal J if A n := J for some positive integer n.The union of all two-sided (or left) nilpotent ideals of S with respect to a two-sided (or left) ideal J is denoted by R(J) (or r(J)).A two-sided (or left) ideal Q is prime (or left prime) if for any two-sided (or left) ideals A, B of S, AB c Q implies that A _= Q or B c Q. We denote by M(J) (or m(J)) the intersection of all two-sided (or left) prime ideals of S containing a two-sided (or left) ideal J.A two-sided (or left) ideal P is completely prime (or left completely prime) if for any a,beS, abeP implies that aeP or be P.We denote by C(A) (or c(A)) the intersection of all two-sided (or left) completely prime ideals of S containing a given subset A.