Some properties of certain simple flat extensions

The ring R such that R[α]∩R[α-1]=R is studied by Ratliff-Mirbagheri ([5]). In [7], they call α anti-integral over R if R[α]∩R[α-1]=R. In [6], the concept of an anti-integral element over R was extended to high degree case. Related papers of birational anti-integral extensions and high degree anti-integral extensions are [1], [2] and [6]. In this paper, we study the simple ring extension A/R dividing to B/R and A/B. In particular, let A=R[α] be a, primitive extension over R (see Definition 2) and put B=R[α]∩R[α-1]. Then the following statements hold. 1) A/B is flat. 2) A/R is flat if and only if B/R is flat. We give the following definition (cf. [6]).