Gr-Semilocal Ring in Graded Algebra

• Gr-semilocal rings are graded rings whose semisimple quotient by the graded Jacobson radical decomposes into gr-simple modules.
• They extend classical semilocality to groupoid-graded structures by employing local units and graded chain conditions.
• Applications include matrix rings, upper triangular matrices, and category algebras, enhancing the analysis of graded module decompositions.

A gr-semilocal ring is a generalization of the classical notion of semilocal rings in the context of groupoid-graded algebra. It is defined as a graded ring $R$ whose semisimple quotient by its graded Jacobson radical consists entirely of gr-simple components, reflecting the semilocality in each local graded piece. This abstraction synthesizes themes from the graded theory of chain conditions, radicals, and module decomposition, and links to the classical Artinian and semilocal frameworks. The concept is pivotal in recent advances involving groupoid gradings, module structure, and categorical generalizations.

1. Foundational Definition and Characterization

For a groupoid-graded ring $R = \bigoplus_{g\in \Gamma} R_g$ (with grading by a groupoid $\Gamma$), the graded Jacobson radical $\operatorname{rad}(R)$ is the intersection of all graded maximal right ideals. $R$ is gr-semilocal if its quotient $R/\operatorname{rad}(R)$ is gr-semisimple; that is, a direct sum of gr-simple modules.

Formally,

$$\text{Gr-semilocal:}\qquad R\ \text{is gr-semilocal} \iff R/\operatorname{rad}(R)\ \text{is gr-semisimple}.$$

For each object idempotent $e \in \Gamma_0$, the "local piece" $R_e \simeq 1_e R 1_e$ is analogously semilocal if $R_e/\operatorname{rad}(R_e)$ is semisimple. A crucial property is

$\operatorname{rad}(R)_e = \operatorname{rad}(R_e)$

ensuring that the local graded structure mirrors the global property.

2. Graded Chain Conditions and Local Units

Chain conditions in the graded setting involve modules or ring components indexed by $\Gamma_0$ satisfying descending or ascending chain conditions (DCC or ACC), leading to the notions of $\Gamma_0$-artinian and $\Gamma_0$-noetherian. Specifically, $R$ is right $\Gamma_0$-artinian if for each $e$, $R_e$ is Artinian in the classical sense. Proposition 7.8 (Cristiano et al., 5 Aug 2025) establishes: $$\text{If } R \text{ is right } \Gamma_0\text{-artinian, then } R \text{ is gr-semilocal}.$$ This condition is verified on each local unit, often without global unitality, leveraging the object-unital structure present in category and groupoid gradings.

3. Graded Jacobson Radical and Gr-Socle Structure

The graded Jacobson radical generalizes the classical radical by intersecting graded maximal (right) ideals. For graded modules $M$, the gr-socle is the sum of all gr-simple submodules. Powers of the graded radical define the Loewy series: $$\operatorname{soc}^{n}_{\mathrm{gr}}(M) = \{m \in M : m \cdot (\operatorname{rad}(R))^n = 0\}$$ capturing "depth" in graded structure akin to classical decomposition into simple layers. In gr-semilocal rings, the interplay of radical powers and socle layers provides refined control over module and ring-theoretic properties.

4. Local and Categorical Generalizations

The definition extends classical semilocal rings and semilocal categories. In particular, for a small preadditive category $\mathcal{C}$, the associated groupoid-graded ring $R_{\mathcal{C}}$ is gr-semilocal if and only if every object of $\mathcal{C}$ has a semilocal endomorphism ring [(Cristiano et al., 5 Aug 2025), Example 7.17]. This categorical perspective enables applications in representation theory and category algebra.

Table: Local-to-Graded Semilocality

Structure	Semilocality Condition	Gr-Semilocal Analogue
Unital ring $A$	$A/\operatorname{rad}(A)$ semisimple	$R_e/\operatorname{rad}(R_e)$ semisimple
Matrix ring $M_I(A)$	$A$ semilocal $\implies M_I(A)$ is semilocal	Graded by $I\times I$, $M_I(A)$ is gr-semilocal
Category algebra $R_\mathcal{C}$	$\operatorname{End}(C)$ semilocal for all $C$	$R_\mathcal{C}$ gr-semilocal

5. Examples and Applications

Explicit constructions illustrate the flexibility of the gr-semilocal concept:

• Matrix rings: If $A$ is semilocal (unital), then $M_I(A)$ with groupoid grading is gr-semilocal: $$\operatorname{rad}(M_I(A)) = M_I(\operatorname{rad}(A))$$ [(Cristiano et al., 5 Aug 2025), Proposition 7.7].
• Upper triangular matrices: For $UT_I(A)$ with $I$ partially ordered, $\operatorname{rad}(UT_I(A))$ consists of matrices with diagonal entries in $\operatorname{rad}(A)$. If $A$ is semilocal, $UT_I(A)$ is gr-semilocal [(Cristiano et al., 5 Aug 2025), Proposition 7.8].
• Category algebras: In preadditive categories, semilocality of object endomorphism rings determines gr-semilocality of the associated algebra.

6. Connections to Classical and Graded Module Theory

Gr-semilocality is a bridge between traditional module theory and its graded/categorical extensions. It harmonizes chain conditions, radical theory, ideal decomposition, and the structure of graded modules. In contexts such as direct-sum decompositions, as studied in (Campanini et al., 17 Apr 2025), gr-semilocal endomorphism rings exhibit module-theoretic properties paralleling classical cases, with block-diagonal decompositions governed by invariants indexed by graded components.

7. Significance and Research Directions

The concept of gr-semilocal rings is central to the development of graded ring theory, allowing the extension of semilocal and Artinian classification to contexts lacking global units but possessing rich local and categorical gradings. It informs ongoing research in graded representation theory, module categories, and algebraic geometry by providing a uniform framework for decomposing and analyzing modules over groupoid-graded structures.

In summary, gr-semilocal rings are essential in understanding local behavior in graded algebraic systems, unifying semilocality, chain conditions, radical theory, and categorical extensions into a coherent generalization with wide-ranging applications and connections to classical results.