Graded Jacobson Radical in Groupoid-Graded Rings

• Graded Jacobson radical is a concept that generalizes the classical radical by capturing maximal superfluous structures in graded rings and modules.
• It exhibits a componentwise structure where the radical localizes along each object in a groupoid, reflecting the simplicity of graded components.
• The theory supports graded chain conditions and semilocality, extending results like Nakayama's lemma and paving the way for further graded module research.

The graded Jacobson radical is a concept that generalizes the classical Jacobson radical to the setting of rings and modules equipped with gradings, including groupoid gradings. It encodes the "maximality" of superfluous or quasi-regular behavior in the sense of graded module and ideal theory, while respecting the grading structure. In groupoid-graded settings, the radical exhibits localization along the object set and reflects the module-theoretic and ring-theoretic simplicity in a highly structured, component-wise manner.

1. Definitions in Groupoid-Graded Contexts

Let $\Gamma$ be a groupoid with object set $\Gamma_0$, and let $R$ be a $\Gamma$-graded ring, meaning $R = \bigoplus_{\gamma \in \Gamma} R_{\gamma}$, with object-unitality (each component $R_e$ for $e \in \Gamma_0$ is unital with local identity $1_e$). For a graded right $R$-module $M = \bigoplus_{\gamma \in \Gamma} M_\gamma$, the graded Jacobson radical of $M$ is defined as the intersection of all gr-maximal graded submodules,

$$\mathrm{rad}(M) = \bigcap \{\, N \subseteq M : N\ \text{is a gr-maximal graded submodule of}\ M \,\}.$$

Equivalently, via the philosophy of superfluousness, $\mathrm{rad}(M)$ is the intersection of all kernels of homomorphisms from $M$ to graded simple modules,

$$\mathrm{rad}(M) = \bigcap \{\, \ker(f) : f \in \mathrm{Hom}_{\mathrm{gr}}(M, S),\ S\ \text{gr-simple} \,\}.$$

The gr-socle of $M$ is the sum of all gr-simple graded submodules.

For a groupoid-graded ring $R$, the graded Jacobson radical is

$$\mathrm{rad}(R) := \mathrm{rad}(R_R) = \mathrm{rad}(_R R) = \bigcap \{\, \mathfrak{m} \subseteq R : \mathfrak{m}\ \text{is a gr-maximal graded right ideal} \,\}.$$

A homogeneous element $a \in R_{\gamma}$ is in the radical if and only if, for every gr-simple module $S$, $Sa = 0$, or (equivalently) for all $x \in R_{\gamma^{-1}}$,

$$1_{r(\gamma)} - a x \in \mathcal{U}(R_{r(\gamma)}),$$

where $r(\gamma)$ is the range (object) of $\gamma$.

2. Componentwise Structure and Localization

A core property in the groupoid-graded theory is the localization of the Jacobson radical along the objects of the grading groupoid. If $R$ is object-unital, then

$$\mathrm{rad}(R) = \bigoplus_{e \in \Gamma_0} \mathrm{rad}(R_e),$$

with $R_e = 1_e R 1_e$ for $e \in \Gamma_0$. Explicitly, for each object $e$, the $e$-component of the graded radical coincides with the ordinary Jacobson radical of the $e$-component subring: $\mathrm{rad}(R)_e = \mathrm{rad}(1_e R 1_e).$ This structure allows for the transfer of classical radical theory to the graded/groupoid context in a componentwise manner.

3. Equivalent Characterizations

Several equivalent criteria for membership in the graded Jacobson radical of a groupoid-graded ring $R$ are established:

• A homogeneous element $a \in R_{\gamma}$ satisfies $a \in \mathrm{rad}(R)$ if and only if $Sa = 0$ for every graded-simple right $R$-module $S$.
• For any $x \in R_{\gamma^{-1}}$, $1_{r(\gamma)} - a x$ is invertible in $R_{r(\gamma)}$.
• $a R$ is a gr-superfluous submodule in $R_R$.
• $a$ belongs to all graded maximal right ideals of $R$. These characterizations generalize the classic "quasi-regular" and "superfluous" criteria to the graded setting, now with respect to the lattice of graded ideals or modules.

4. Chain Conditions and Semilocality in Graded Settings

The paper introduces graded chain conditions for modules and rings with groupoid gradings. A module is $\Gamma_0$-artinian if it has no infinite descending chain of graded submodules, and $\Gamma_0$-noetherian if there are no infinite ascending chains. Many theorems analogous to the classical artinian/noetherian theory hold, though examples are given where $\Gamma_0$-artinian rings need not be $\Gamma_0$-noetherian.

A graded semilocal ("gr-semilocal") ring is defined as a groupoid-graded ring whose semisimple quotient $R/\mathrm{rad}(R)$ is gr-semisimple, generalizing both semilocal rings and semilocal small categories. For gr-semilocal rings, key properties include

$\mathrm{rad}(M) = M \cdot \mathrm{rad}(R),$

and the existence of a well-behaved Loewy (socle) series,

$\soc_{\mathrm{gr}}^n(M)= \{\, m\in M : m \cdot [\mathrm{rad}(R)]^n = 0\,\}.$

5. Concrete Examples and Applications

Several archetypal examples illustrate the groupoid-graded radical theory:

• For the upper triangular matrix ring $UT_I(A)$ over a unital ring $A$, graded by $I\times I$, the graded Jacobson radical is

$$\mathrm{rad}(UT_I(A)) = \{ (a_{ij}) \in UT_I(A) : \forall\, i \in I,\, a_{ii} \in \mathrm{rad}(A) \},$$

and for the full matrix ring $M_I(A)$,

$\mathrm{rad}(M_I(A)) = M_I(\mathrm{rad}(A)).$

These formulas demonstrate the localization of the radical to the diagonal components dictated by the groupoid structure.

The graded version of Nakayama's lemma is proven for groupoid-graded rings: if $J \subseteq \mathrm{rad}(R)$, for any finitely (or $\Gamma_0$-) generated graded right module $M$, $MJ = M$ implies $M = 0$.

6. Comparison with Classical Radical Theory

All central features of the classical Jacobson radical extend to the groupoid-graded context, but with more refined structure:

• The radical is still the intersection of (graded) maximal right ideals, but these ideals are now considered within the graded module/ideal lattice.
• The radical may be computed componentwise via the grading groupoid's objects.
• Simplicity, primitive ideals, and maximality are all interpreted through the graded module theory over the groupoid.

7. Significance and Further Directions

The development of the graded Jacobson radical for groupoid-graded rings and modules provides a framework that unifies and extends the classical radical theory. It accommodates both algebraic and categorical gradings, supports localization phenomena, and ensures compatibility with graded semilocality and graded module-theoretic chain conditions.

These results serve as a foundation for further work on graded injective and projective modules, graded semiperfect and semilocal rings, and investigations into spectral and decomposition theories for objects in groupoid-graded settings. The ability to express many structural properties via direct sums over the object set opens avenues for decomposition, classification, and duality theory in settings relevant to both algebraic and categorical representation theory.