Regularity of Noetherian local rings

Let (R,m) be a Noetherian local ring of prime characteristic p. In this paper, as an extension of the concept of pn-basis, we introduce the notion of m-adic pn-basis, and we show that R/Rpn has an m-adic pn-basis for every n (n=1,2,…) if and only if R is a regular local ring. For any Noetherian local ring, further we introduce the concept of m-adic free graded algebra and we give a regularity criterion in terms of the higher differential algebra, and others.