[Paper Review] New characterizations for core inverses in rings with involution

Sanzhang Xu, Jianlong Chen+1 more|arXiv (Cornell University)|Dec 26, 2015

Matrix Theory and Algorithms11 references3 citations

TL;DR

This paper provides new three-equation characterizations for core inverses in rings with involution, establishing that an element $ a $ is core invertible with inverse $ b $ if and only if $ (ab)^* = ab $, $ ba^2 = a $, and $ ab^2 = b $. It further proves that a group invertible element is core invertible if and only if it is also a {1,3}-inverse, and derives explicit formulas for the core inverse of the sum of two core invertible elements under specific orthogonality conditions.

ABSTRACT

The core inverse for a complex matrix was introduced by Baksalary and Trenkler. Rakić, Dinčić and Djordjević generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible, in this paper, we will answer this question. We will use three equations to characterize the core inverse of an element. That is, let $a, b\in R$, then $a\in R^{ iny extcircled{ iny\#}}$ with $a^{ iny extcircled{ iny\#}}=b$ if and only if $(ab)^{\ast}=ab$, $ba^{2}=a$ and $ab^{2}=b$. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

Motivation & Objective

• To determine when a group invertible element in a *-ring is core invertible.
• To provide a minimal three-equation characterization of the core inverse, reducing the five-equation system from prior work.
• To investigate the additive properties of core invertible elements and derive explicit formulas for the core inverse of their sum.
• To extend known results on core inverses from complex matrices to general rings with involution.
• To clarify the relationship between core invertibility, group invertibility, and {1,3}-invertibility in *-rings.

Proposed method

• Proposes a three-equation characterization: $ (ab)^* = ab $, $ ba^2 = a $, and $ ab^2 = b $, to define the core inverse of $ a $ with $ b $ as its inverse.
• Uses the equivalence between core invertibility and the intersection of group invertibility and {1,3}-invertibility to derive necessary and sufficient conditions.
• Applies ring-theoretic tools, including involution properties, idempotents, and annihilator conditions, to analyze the structure of core invertible elements.
• Derives the formula $ (a+b)^{ iny{ iny ext{ extsterling}}}=b^{ ext{ extpi}}a^{ iny{ iny ext{ extsterling}}}+b^{ iny{ iny ext{ extsterling}}} $ under conditions $ ab = 0 $, $ a^*b = 0 $, and $ a,b $ core invertible.
• Utilizes the concept of $ b^{ ext{ extpi}} = 1 - b^{ iny{ iny ext{ extsterling}}}b $, the spectral idempotent associated with the group inverse, to express the sum's core inverse.
• Validates results through counterexamples to show the necessity of orthogonality conditions in sum formulas.

Experimental results

Research questions

• RQ1 When is a group invertible element in a *-ring also core invertible?
• RQ2 Can the core inverse of an element be characterized by only three equations rather than five?
• RQ3 Under what conditions is the sum of two core invertible elements itself core invertible?
• RQ4 What is the explicit formula for the core inverse of the sum of two core invertible elements satisfying $ ab = 0 $ and $ a^*b = 0 $?
• RQ5 How do the core inverse and dual core inverse behave under additive conditions, and what are the corresponding sum formulas?

Key findings

• An element $ a $ is core invertible with inverse $ b $ if and only if $ (ab)^* = ab $, $ ba^2 = a $, and $ ab^2 = b $, providing a minimal three-equation characterization.
• A group invertible element $ a $ is core invertible if and only if it is also a {1,3}-inverse, i.e., $ a otin R^{ iny{ iny ext{ extsterling}}} $ if and only if $ a otin R^{ ext{#}} igcap R^{ ext{\{1,3\}}} $.
• If $ a $ and $ b $ are core invertible and satisfy $ ab = 0 $ and $ a^*b = 0 $, then $ a + b $ is core invertible with $ (a+b)^{ iny{ iny ext{ extsterling}}} = b^{ ext{ extpi}}a^{ iny{ iny ext{ extsterling}}} + b^{ iny{ iny ext{ extsterling}}} $, where $ b^{ ext{ extpi}} = 1 - b^{ iny{ iny ext{ extsterling}}}b $.
• The condition $ ab = 0 $ in the sum formula is necessary, as shown by counterexamples where $ ab \neq 0 $ leads to failure of core invertibility of $ a + b $.
• The dual core inverse of the sum $ a + b $ under $ ab = 0 $ and $ ab^* = 0 $ satisfies $ (a+b)_{ iny{ iny ext{ extsterling}}} = a_{ iny{ iny ext{ extsterling}}} + b_{ iny{ iny ext{ extsterling}}}a^{ ext{ extpi}} $, with $ a^{ ext{ extpi}} = 1 - aa_{ iny{ iny ext{ extsterling}}} $.
• The results generalize known matrix-theoretic results on core inverses to arbitrary rings with involution, establishing broader applicability.