Relatively Orthocomplemented Skew Nearlattices in Rickart Rings

Abstract

Abstract. A class of (right) Rickart rings, called strong, is isolated. In particular, every Rickart *-ring is strong. It is shown in the paper that every strong Rickart ring R admits a binary operation which turns R into a right normal band having an upper bound property with respect to its natural order ≤; such bands are known as right normal skew nearlattices. The poset pR, ≤q is relatively orthocomplemented; in particular, every initial segment in it is orthomodular.The order ≤ is actually a version of the so called right-star order. The one-sided star orders are well-investigated for matrices and recently have been generalized to bounded linear Hilbert space operators and to abstract Rickart *-rings. The paper demonstrates that they can successfully be treated also in Rickart rings without involution.

Strongly Rickart *-rings

“…A is a * -abelian). Finally, the author in [7] introduced the strong Rickart ring as a generalization to strongly Rickart Ring and to a Rickart * -ring. Proof.…”