Abstract
Let k be an algebraically closed field of prime characteristic p. We shall investigate the relation between the existence of a regular subring R′ of R = k[x, y] containing R p = k[x p , y p ] and the existence of a p-basis of R (resp., R′) over R′ (resp., R p ). Moreover, we give examples such that the existence of a p-basis of R over R′ is dependent on characteristic p and R′ is not a polynomial ring but a regular factorial ring, and we show that R has a p-basis over R p [f] (f ∈ R − R p ) if deg f ≤ 3 and R p [f] is regular.
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Communicated by I. Swanson.