A remark on arithmetical rings

ΦA,B := {I | I is non-zero finitely generated, A ⊆ I, I + B = R} . He showed, [3, Theorem 5], that M is the smallest element in ΦA,B if and only if the following conditions are satisfied: ( i) A ⊆ M , ( ii) M + B = R and ( iii) if S is a non-zero a finitely generated ideal in R such that AM−1 ⊆ S and S + B = R, then R = S. He also proved, [3, Corollary to Theorem 4], that the smallest element always exists if R is a Dedekind domain. In this short note we generalize these two results to arithmetical rings. Recall that a ring R is called an arithmetical ring if every finitely generated ideal in R is multiplication. An ideal A in R is multiplication if for every ideal B ⊆ A, there exists an ideal C in R such that B = CA, [2]. Note that C ⊆ [B : A], and hence B = CA ⊆ [B : A]A ⊆ B,