Quasicommutative weakly primary semigroups
František Kmeť · Czech digital mathematics library · 1982
QUASICOMMUTATTVE WEAKLY PRIMARY SEMIGROUPS FRANTlSEK KMEtLet S be a quasicommutative semigroup, i.e. a semigroup with ab = b'a for all elements a, b of S and for some positive integer r=r(a, b).Quasicommutative semigroup have been studied by N. P. Mukherjee in [5].M. Satyanarayanain[6] studied primary ideals in commutative semigroups and commutative primary semigroups.An ideal A of a commutative semigroup is called primary if for x, y e S, xy e A and x^A imply y" e A for some positive integer n.A commutative semigroup is called primary if every ideal of the semigroup is primary.An ideal P of an arbitrary semigroup is called prime (completely prime) if for any two ideals A, B (elements a, b) of the semigroup, AB<=P (abeP) implies that either AcP or BcP (aeP or beP).In the present paper we shall study quasicommutative weakly primary ideals and weakly primary semigroups.An ideal of a quasicommutative semigroup S is said to be weakly primary if x, yeS, xyeQ implies that either x m e Q or y" e Q for some positive integers m, n.An analogous notion can be found in [4, 247] (for subsemigroups).A quasicommutative semigroup S is said to be weakly primary if every ideal of S is weakly primary.Evidently every completely prime ideal is weakly primary.Also every primary ideal is weakly primary.Conversely, a weakly primary ideal need not be primary.This is shown on the following example.Let Si = {0, 4, 6} be the semigroup with the multiplication by mod 12.The all ideals of Si are (0), ( 4), (6), Si.The primary ideals of Si are (4), (6) and Si.The ideal (0) is not primary since 6.4 € (0), 6 ^ (0) but for any positive integer n we have 4" = 4 ^ (0).However, the ideal (0) is weakly primary.It is known (H.Lai [3, Theorem]) that in a quasicommutative semigroup S with an ideal J the set of all nilpotent elements of S with respect to J (i.e. the set of all xeS such that x" e J for some positive integer n) is equal to the radicals R(J) of Schwarz, M(J) of McCoy, L(J) of Sevrin, C(J) of Luh, R*(J) of Clifford, respectively.For example the Schwarz radical R(J) is the union of all nilpotent ideals of S with respect to / (i.e. the union of all ideals I of S with I" c.J for some