D-core Algebra: Dual Core Inverses in *-Rings

• D-core Algebra is a framework that generalizes dual core inverses in *-rings, defined by unique inner inverses with specific adjoint and range constraints.
• It employs idempotent decompositions and operator-theoretic techniques to establish existence criteria and structural properties of elements in *-rings.
• The theory bridges connections with group, Moore–Penrose, and (b,c)-inverses, offering practical insights for analyzing singular operators and matrix factorizations.

The D-core algebra centers on the concept of the dual core (D-core) inverse in the context of *-rings, which are associative rings equipped with an involution operation. Extending developments from the theory of generalized matrix inverses, the D-core inverse generalizes the notion to arbitrary *-rings, providing a unique inner inverse with specific range and adjoint-related constraints. Its existence, algebraic structure, operator-theoretic implications, and connections to other classes of generalized inverses (such as group, Moore–Penrose, and core inverses) have significant implications for abstract algebra and operator theory (Rakić et al., 2014).

1. Generalized Inverses in *-Rings: Preliminaries

Let $R$ be a *-ring (an associative ring equipped with an involution $*$). Generalized inverses in $R$ are framed by reflexivity and idempotent properties:

• An element $a \in R$ is inner generalized invertible (von Neumann regular) if there exists $x\in R$ such that $a x a = a$.
• Outer generalized invertibility holds if $y\in R$ satisfies $y a y = y$.
• Reflexive generalized inverses require $x$ to be both an inner and an outer inverse for $a$.

Principal ideals and annihilators introduce further structure:

• $aR$ (right principal ideal), $Ra$ (left principal ideal)
• $a^\circ = \{ x: ax=0 \}$ and $^\circ a = \{ x: xa=0 \}$ (right and left annihilators)

The classical group inverse ($a^\#$) and Moore–Penrose inverse ($a^\dagger$) are uniquely defined for $a$ in $R$ subject to systems of algebraic equations involving $a$, $x$ and the involution $*$.

2. Definition and Characterizations of the D-Core Inverse

The D-core (dual core) inverse for $a\in R$ is denoted $a^g$. Its definition balances the range conditions of the adjoint:

An element $x\in R$ is a dual core inverse of $a$ ($x=a^g$) if:

• $a x a = a$
• $xR = a^* R$
• $R x = R a$

Equivalently: $x^\circ = (a^*)^\circ$ and $^\circ x = a^\circ$. $a^g$ is a reflexive inner inverse, directly paralleling the core inverse but exchanging the roles of $a$ and $a^*$ in the ideal conditions.

3. Existence Criteria and Idempotent Decompositions

The existence of $a^g$ is equivalent to $a$ being inner-invertible and the annihilator conditions aligning with unique idempotent elements. These idempotents enable canonical decompositions:

• There exist idempotents $q, r \in R$ (with $r$ self-adjoint) such that
  • $qR = aR$, $Rq = Ra$, $Rr = Ra$
• Block decomposition:

$$a = \begin{bmatrix} a_{11} & 0 \ 0 & 0 \end{bmatrix}_{q \times r}, \quad a^g = \begin{bmatrix} a_{11}^{-1} & 0 \ 0 & 0 \end{bmatrix}_{r \times q}$$

where $a_{11}$ is invertible in the corner ring $rRq$.

The conditions are structurally analogous to those characterizing group and Moore–Penrose inverses, but with the critical differentiation in the ideals generated.

4. Equivalent Characterizations and Equational Systems

Several characterizations of the D-core inverse are equivalent:

1. $x = a^g$, that is, the dual core inverse exists.
2. $a x a = a$, $x^\circ = (a^*)^\circ$, $^\circ x = a^\circ$.
3. $a x a = a$, $x a x = x$, $(x a)^* = x a$, $a^2 x = a$, $x^2 a = x$.
4. Existence of the aforementioned idempotents in the corresponding 2×2 block decomposition.
5. If $a$ is inner-invertible and the annihilator conditions coincide as above, then $x = r a^{(1)} q$, for an arbitrary inner inverse $a^{(1)}$.

The uniqueness and explicit construction via idempotents and corner subrings offer powerful tools for analyzing the algebraic structure.

5. Comparison with Other Generalized Inverses

The D-core inverse locates itself between the group inverse $a^\#$ and the Moore–Penrose inverse $a^\dagger$, both algebraically and in operator-theoretic settings:

• If $a^g$ exists, so does $a^\#$, with $a^\# = a^\ell a^g$.
• If $a^\ell$ or $a^g$ exists, $ind(a) \leq 1$, guaranteeing the existence of $a^\dagger$ under mild supplementary conditions (e.g., in Rickart *-rings or for operators with closed range).
• In operator block-decompositions, $a$, $a^\#$, $a^\dagger$, and $a^g$ occupy distinct but structurally mirrored subblocks, as formalized in full matrix and operator settings.

EP elements—those for which the group and Moore–Penrose inverses coincide—are characterized by $a^\ell = a^g$, $a^\# = a^\dagger$, and further by commuting idempotents and various mixed equations.

6. Inverses Along an Element and (b,c)-Inverses

The D-core inverse can be understood within broader classes of generalized inverses:

• Inverse along an element: $a^g$ exists if and only if $a$ is invertible along $a^* a$; the inverse along $a^* a$ coincides with $a^g$.
• (b,c)-inverse: $a^g$ exists if and only if $a$ admits a $(b=a^*, c=a)$-inverse, with the unique $(a^*, a)$-inverse being $a^g$.

This situates the D-core inverse within a unifying framework for generalized invertibility, linking it to classical constructions and more recent generalizations.

7. Operator-Theoretic Context and Applications

In the algebra $B(H)$ of bounded linear operators on a Hilbert space $H$:

• $A\in B(H)$ has a dual core inverse $A^g$ if and only if $ind(A)\leq 1$, i.e., if $R(A)$ is closed and $R(A) = R(A^2)$, $N(A) = N(A^2)$.
• The operators decompose via projections (idempotents) onto $R(A)$ and $R(A^*)$, allowing the realization of $A^g$ as the inverse of the corner operator $A_{11}:R(A)\to R(A^*)$.
• This algebraic structure has immediate implications for the analysis of differential equations with singular operators and for matrix factorizations.

Potential extensions involve Banach -algebras, C-module categories, and weighted or constrained variants of the D-core inverse. Open problems include the generalization to rings lacking involution and further structural classification within and beyond operator theory (Rakić et al., 2014).