Extending partial orders on rings

Abstract

For a commutative ring with identity, it is consistently L^∗ if and only if it is consistently O^∗. This result is generalized to noncommutative rings with identity in the section 2. In the section 3, it is shown that for any abelian group G and integral domain D, the group ring D[G] is L^∗ if and only if it is O^∗.

Mathematics Subject Classification (2010):

• Primary: 06F25

Key words:

• Consistently L^∗-ring
• consistently O^∗-ring
• left division closed
• L^∗-ring
• O^∗-ring