Natural partial order on rings with involution

Peer review under responsibility of Kalasalingam University.

Abstract

In this paper, we introduce a partial order on rings with involution, which is a generalization of the partial order on the set of projections in a Rickart $\ast$-ring. We prove that a $\ast$-ring with the natural partial order forms a sectionally semi-complemented poset. It is proved that every interval $[0,x]$ forms a Boolean algebra in case of abelian Rickart $\ast$-rings. The concepts of generalized comparability $\left(GC\right)$ and partial comparability $\left(PC\right)$ are extended to involve all the elements of a $\ast$-ring. Further, it is proved that these concepts are equivalent in finite abelian Rickart $\ast$-rings.

Keywords:

• $\ast$-ring
• Partial order
• Generalized comparability
• Partial comparability

Notes

Peer review under responsibility of Kalasalingam University.