TITLE:
Some Results on Partially Ordered Rings
AUTHORS:
Jingjing Ma
KEYWORDS:
Algebraic, Archimedean, Infinite Prime, Integral Domain, Maximal Partial Order, Negative Square, Symmetric Partial Order, Total Order
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.12,
December
23,
2025
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.