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Primary Decomposition in A r-Semigroup

Dept. of Mathematics,V S R & N V R College, Tenali, A.P. India. Email: 1raoag1967@gmail.com, 2anjaneyulu.addala@gmail.com, 3dmrmaths@gmail.com

ABSTRACT

In this paper the terms P-primary, primary decomposition of a r-ideal, reduced primary decomposition of a  r-ideal in a  r-semigroup S areintroduced.  If A1, A2, …An are P-primary  r-ideals in a  r-semigroup S, then it is proved that  is also a P-primary r-ideal.  If a  r-ideal A in a  r-semigroup S has a primary decomposition, then  it is proved that A has a reduced primary decomposition.  Further it is proved that every Γ-ideal in a (left, right) duo noetherian
Γ-semigroup S has a reduced (right, left) primary decomposition.  If A and B are two Γ-ideals in a Γ-semigroup S, then it is proved that   Al (B) = { x S : < x >ΓB A} and
Ar (B) = { x
S : B Γ< x > A} are Γ-ideals of S containing A.   Further it is proved that (1) if A is a left primary  r-ideal of a  r-semigroup S, then Al (B) is a left primary  r-ideal,  (2) if A is a right primary  r-ideal of a  r-semigroup S, then Ar (B) is a right primary  r-ideal.   It is proved that if Q is a P-primary  r-ideal and if A P, then Ql (A) = Qr (A) = Q and also if A P and A Q, then √( Ql (A)) = √( Qr (A)) = √Q.  If A1, A2,….. An, B are  r-ideals of a  r-semigroup S, then it is proved that.   Further if a  r-ideal A in a  r-semigroup S has two reduced (one sided) primary decompositions; A = A1∩A2∩……∩Ak = B1∩B2∩…..Bs, where Ai is Pi -primary and Bj is Qj-primary, then it is proved that k = s and after reindexing if necessary
Pi = Qi for i = 1, 2, …,k.

International eJournal of Mathematical Sciences, Technology and Humanities

Volume 2, Issue 3, Pages:  466 - 479