PRIMARY DECOMPOSITION IN TERNARY SEMIGROUPS

Abstract

In this paper, the terms P- primary ideal, primary decomposition, noetherian
ternary semigroup. are introduced. It is proved that, if A1, A2, . . . . An are Pprimary
ideals in a ternary semigroup T, then is also a P-primary ideal. It is also proved that, if an ideal A in a ternary semigroup T has a primary decomposition, then A has a reduced primary decomposition, Further it is proved that Every ternary ideal in a commutative neotherion ternary semigroup T has a reduced primary decomposition. It is proved that, if A ia a left / lateral / right primary ideal of a terary semigroup T, then is a left /lateral/ right primary ternaryideal. It is also proved that, Q is a P-primary ternary ideal and if A ⊈P, then = = = Q, and also if A⊆ P and A ⊈ Q, then = = = and if A1, A2, A3, . . . . An, B are ternary ideals of a ternary semigroup T, then )l(B) =)l(B). Finally it is proved that, in a ternary semigroup T, an ideal A has two reduced (one sided) primary decomposition A = A1 ∩ A2 ∩……∩ Ak = B1∩B2 ∩…..∩ Br where Ai is pi- primary and Bj is Qj- primary. Then k = r and after re-indexing if necessary, Pi = Qi for i=1,2,…k. Further if each Pi is
an isolated prime, then Ai = Bi for i = 1, 2, …., n.