On a p-Basis and a Regular Subring of a Polynomial Ring in Two Variables

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.

Key Words:

• Derivation
• Factorial
• p-Basis
• Polynomial ring
• Regular

2000 Mathematics Subject Classification:

• Primary 13B99
• Secondary 14A10

Notes

Communicated by I. Swanson.