Some Results on Semi groups when f-Prime Ideals are Maximal

Abstract

In this research paper it is verified that S becomes f-primary if proper f-prime ideals are maximal in S. It is verified that for S being quasi-commutative, right cancellative and S is either a f-primary (or) S being a semi group with f-semi primary ideals are f-primary then proper f-prime ideals are maximal. We proved that S being cancellative and commutative with either S is f-primary (or) an ideal Q is f - primary in S ⟺ r_f(Q) is a f-prime ideal, and therefore the proper f-prime ideals in S are maximal. It is verified that for S is quasi-commutative and right cancellative having identity, then these statements are equivalent. (1) proper f-prime ideals are maximal. (2) S being f-primary (3) f-semi primary ideals are f-primary. (4) If g & h are not the units in S, then ∃ n, m∈N ∋ gⁿ = hs and h^m = gr for any s, r ∈ S. It is shown that if S is a quasi-commutative and right cancellative without identity then these conditions are equivalent (1) S being f- primary (2) f-semi primary ideals are f- primary (3) There are no proper f-prime ideals in S (4) g and h are not units in S, then ∃ n, m∈N ∋ gⁿ = hs and h^m = gr for any s, r ∈ S. It is verified that for S being quasi-commutative and right cancellative then these statements (1) S is f-primary (2) f-semi primary ideals are f-primary (3) proper f-prime ideals are maximal, are equivalent.