Some Results on Partially Ordered Rings

Abstract

The paper presents some results on partially ordered rings. In section 2, it is shown that Archimedean directed maximal partial orders on integral domains must be total orders. Section 3 presents a direct proof that fields not algebraic over ℚ admit directed partial orders. Section 4 mainly considers the connection between symmetric partial orders, directed maximal partial orders and full infinite primes for commutative rings that are algebraic over ℤ . In particular, it is shown that in commutative rings that are algebraic over ℤ and do not contain nonzero nilpotent elements, the symmetric partial orders and full infinite primes are in one-to-one correspondence.