| Let R be a commutative ring with identity. A multiplicative subsemigroup S of (R,·) is called an annihilator-semigroup for R if for each r ∈ R there exists a unique s ∈ S with 0 : r = 0 : s. If R admits an annihilator-semigroup, R is called an annihilator-semigroup ring. The quotient monoid R/≡ where a ≡ b ⇔ 0 : a = 0 : b is called the annihilator congruence semigroup of R. If S is an annihilator-semigroup for R, then S ≈ R/≡. In this thesis we continue the investigation of annihilator-semigroups and annihilator congruence semigroups began by Anderson and Camillo.;Let P(R) be the monoid of principal fractional ideals of R. Define the equivalence relation ≈ on P(R) by Ra ≈ Rb if and only if Ra and Rb are isomophic as R-modules. We show that R/≡ and P(R)/≈ are isomophic as monoids.;The set F¯(R) of R-submodules of T(R), the total quotient ring of R, is a partially ordered commutative monoid with identity R and zero 0 under the usual product IJ = {Sigma ialphajalpha| ialpha ∈ I, j alpha ∈ J}. We investigate when F¯ (R) or certain of its submonoids are finitely generated. In particular, we consider the finite generation of the submonoids P(R) ⊆ F*(R) ⊆ F(R) ⊆ F¯(R) where P(R) (resp., F*( R), F(R)) is the set of principal fractional ideals (resp., finitely generated fractional ideals, fractional ideals) of R and of their positive cones P +(R) ⊆ F*+ (R) ⊆ F+( R) = F¯+(R) which consist of the principal ideals, finitely generated ideals, and ideals of R, respectively. We show that P( R) is finitely generated if and only if P( R¯) (R¯ the integral closure of R) is finitely generated and R¯/[R :R¯] is finite. Moreover, R¯ is a finite direct product of finite local rings, SPIRs, Bezout domains D with P(D) finitely generated, and special Bezout rings S (S is a Bezout ring with a unique minimal prime P, SP is an SPIR, and P(S|P) is finitely generated). Also, P(R) is finitely generated if and only if F*(R), the monoid of finitely generated fractional ideals of R, is finitely generated. We show that the monoid F(R) of fractional ideals of R is finitely generated if and only if the monoid F¯(R) of R-submodules of the total quotient ring of R is finitely generated and characterize the rings for which this is the case. |
