In the first chapter, we discuss the fully graded algebras over discrete valuation ring with an algebraically closed residue field of characteristic p.

We give a new characterization of weakly primary rings with identity and conditions that a weakly primary domain is a valuation ring or quasi-valuation ring.

In this paper, we introduce the notion of M-valuation systems for a commutative ring, and investigate the M-valuations from the inside of a commutative ring.