On the dual of an ideal of a g-monoid

Abstract

~on S×S by (s1,t1)~(s2,t2) if s1+t2=s2+t1. We denote the equivalence class of (s,t) under~by s-t. Let G={s-t|s,t∈S}be the set of equivalence classes. Then G is a monoid with identity 0=s-s with each s∈S under the additive operation (s1-t1)+(s2-t2)=(s1+s2)-(t1+t2). Furthermore, each element s-t of G has the converse t-s and then G is a torsion-free Abelian group. Evidently S is a submonoid of the group G. The group G is called the quotient group of S. The quotient group of a monoid S is often denoted by q(S). Let S be a g-monoid with quotient group G. A subset I of G is called a fractional ideal of S if S+I⊆I and s+I⊆S for some element s∈S. A subset I of S is called an integral ideal of S if I+S⊆I. We shall denote the set of fractional ideals of S by F(S). For each element x of G, the set x+S={x+s|s∈S} is a fractional ideal of S and is called a principal ideal of S. The principal ideal x+S is simply denoted by (x).