Core partial order in rings with involution

Abstract

Let R be a unital ring with involution. Several characterizations and properties of core partial order are given. In particular, we investigate the reverse order law (ab) # = b # a # for two core invertible elements a, b ∈ R. Some relationships between core partial order and other partial orders are obtained.

Partial Orders Based on the CS Decomposition

“…[22, Theorem 3.2] Let A, B ∈ C n×n be two core invertible matrices. ThenA # ≤ B if and only if A * ≤ B and B # AA # = A #…”