Jacobson Localization in Noncommutative Rings

• Jacobson Localization is a noncommutative localization mechanism defined through the Jacobson radical and quasi-regularity, which constructs rational closures in I-adically complete overrings.
• It embeds rings into their completions and adjoins Jacobson quasi-inverses, linking the process to Cohn, Leavitt, and Gabriel localizations with applications in free and Fox algebras.
• This framework unifies various localization methods while clarifying limitations in differential Ore extensions and profinite algebra contexts.

Jacobson localization denotes a family of localization procedures organized around the Jacobson radical, Jacobson quasi-regularity, or the Jacobson spectrum. In the noncommutative setting represented by recent work, the term has both a broad structural meaning and a more specific constructive meaning. Broadly, it refers to localization constructions that respect the Jacobson radical or invert sets of elements avoiding it, together with the study of how Jacobson radicals behave under Ore extensions, profinite duality, and related ring-theoretic operations (Smoktunowicz et al., 2013). More specifically, Jacobson localization can be realized by embedding a ring $R$ into an $I$-adically complete overring $\hat{R}$, identifying an ideal $I\subseteq J(\hat{R})$, and then closing $R$ under Jacobson quasi-inverses of quasi-regular elements; in the free and skew-free settings, this rational closure coincides with a Cohn localization by matrices of the form $1+Q$ with $Q\in M_I$ (Anh, 3 Oct 2025).

1. Jacobson quasi-regularity and the basic localization mechanism

A central algebraic notion is quasi-regularity in the sense of Jacobson. An element $a$ is quasi-regular if there exists $b$ such that

$a + b + ab = 0 = a + b + ba.$

Equivalently, an element $I$0 in a ring $I$1 is quasi-regular if $I$2 is invertible in $I$3. The element $I$4 is the Jacobson quasi-inverse $I$5, and one has

$I$6

(Anh, 3 Oct 2025).

The constructive setup begins with a two-sided ideal $I$7 satisfying

$I$8

Then the $I$9-adic topology on $\hat{R}$0 is Hausdorff, and the completion is

$\hat{R}$1

If $\hat{R}$2, the series $\hat{R}$3 converges in $\hat{R}$4, and

$\hat{R}$5

is a Jacobson quasi-inverse. Consequently, all elements of $\hat{R}$6 are Jacobson quasi-regular in $\hat{R}$7, and $\hat{R}$8 is a multiplicative subgroup of $\hat{R}$9 (Anh, 3 Oct 2025).

This construction supplies the core mechanism of Jacobson localization: one works inside a complete overring where geometric-series quasi-inverses exist, and then regards localization as the process of adjoining those quasi-inverses to the original ring. A plausible implication is that Jacobson localization is less an inversion of arbitrary denominators than a controlled closure under quasi-inversion dictated by an ideal topology.

2. Rational closure as Jacobson localization

For a ring $I\subseteq J(\hat{R})$0 with a transfinitely nilpotent ideal $I\subseteq J(\hat{R})$1, the rational closure with respect to $I\subseteq J(\hat{R})$2 is defined as the smallest subring of $I\subseteq J(\hat{R})$3 containing $I\subseteq J(\hat{R})$4 and closed under taking quasi-inverses of its quasi-regular elements. Elements of this ring are called rational (Anh, 3 Oct 2025). Concretely, one sets $I\subseteq J(\hat{R})$5, $I\subseteq J(\hat{R})$6, then for each $I\subseteq J(\hat{R})$7 lets $I\subseteq J(\hat{R})$8 be the subring of $I\subseteq J(\hat{R})$9 generated by $R$0 and the subgroup of $R$1 generated by $R$2, with $R$3. The rational closure is

$R$4

(Anh, 3 Oct 2025).

This procedure is idempotent: if $R$5 and $R$6 is the ideal of $R$7 generated by $R$8, then $R$9 and

$1+Q$0

(Anh, 3 Oct 2025). In this sense, rational closure behaves as a localization functor adapted to Jacobson quasi-regularity rather than to central denominators.

The same paper gives a matrix characterization. For any $1+Q$1, any $1+Q$2, and any column vector $1+Q$3 with entries in $1+Q$4, the equation

$1+Q$5

has a unique solution $1+Q$6 whose entries lie in $1+Q$7; similarly for right equations $1+Q$8. Equivalently, the matrices $1+Q$9 with $Q\in M_I$0 are invertible over $Q\in M_I$1, and $Q\in M_I$2 is exactly the set of entries of unique solutions $Q\in M_I$3 in $Q\in M_I$4 of equations

$Q\in M_I$5

(Anh, 3 Oct 2025).

This identifies Jacobson localization with a closure under solvability of linear systems whose coefficients are Jacobson-controlled perturbations of the identity. The paper’s formulation suggests that the Jacobson condition is encoded at the matrix level, not merely at the level of single elements.

3. Relation to Cohn, Leavitt, and Gabriel localizations

A major structural result is that rational closure coincides with a Cohn localization. If $Q\in M_I$6, then the universal $Q\in M_I$7-inverting ring is the Cohn localization

$Q\in M_I$8

For $Q\in M_I$9 equal to the free associative algebra $a$0 or the free group algebra $a$1 on finitely many generators, with augmentation ideal $a$2, one has

$a$3

inside the $a$4-adic completion $a$5 (Anh, 3 Oct 2025). The same identification is stated for the more general situation in which $a$6 is a fir and $a$7 is any proper ideal (Anh, 3 Oct 2025).

The same framework is linked to Leavitt localization. Given generators $a$8, the Leavitt localization is the universal ring obtained by adjoining $a$9 subject to

$b$0

If

$b$1

is a two-sided ideal with trivial left annihilator, then the Leavitt localization $b$2 is isomorphic to the flat epimorphic right ring of quotients $b$3 for the $b$4-adic Gabriel topology (Anh, 3 Oct 2025). The canonical map $b$5 is injective, and $b$6 is free of rank $b$7 over $b$8 (Anh, 3 Oct 2025).

These identifications place Jacobson localization within a localization triangle. First, rational closure gives a Jacobson-type localization via quasi-inverses in the completion. Second, the same ring is a Cohn localization inverting all matrices $b$9 over $a + b + ab = 0 = a + b + ba.$0. Third, when $a + b + ab = 0 = a + b + ba.$1 is generated by a free row, a further Leavitt row-localization appears as a Gabriel localization. This suggests that Jacobson localization is not a rival to matrix or quotient localizations; rather, it furnishes a common framework in which they can coincide.

4. Free algebras, free group algebras, and the Fox algebra

For the free algebra $a + b + ab = 0 = a + b + ba.$2 with augmentation ideal $a + b + ab = 0 = a + b + ba.$3, and the free group algebra $a + b + ab = 0 = a + b + ba.$4 with augmentation ideal $a + b + ab = 0 = a + b + ba.$5, one has

$a + b + ab = 0 = a + b + ba.$6

hence the $a + b + ab = 0 = a + b + ba.$7-adic topology is Hausdorff; the $a + b + ab = 0 = a + b + ba.$8-adic topology is also Hausdorff (Anh, 3 Oct 2025). There are embeddings

$a + b + ab = 0 = a + b + ba.$9

and $I$00 (Anh, 3 Oct 2025). This gives a concrete ambient ring in which Jacobson quasi-inverses are represented by convergent noncommutative power series.

For a free group $I$01, setting $I$02 yields

$I$03

This ideal is transfinitely nilpotent and free of rank $I$04 as a right $I$05-module, with trivial left annihilator (Anh, 3 Oct 2025). The corresponding ring of right quotients is realized as a Fox algebra. The right Fox derivatives $I$06 form the entries of the inverse column to the row $I$07, and adjoining all Fox derivatives yields the Leavitt localization $I$08, called the Fox algebra of rank $I$09 (Anh, 3 Oct 2025).

In this setting the Fox algebra combines three perspectives: a Cohn/Jacobson localization at the augmentation ideal via rational closure, a Leavitt localization inverting the generating row of the ideal, and a Gabriel localization as a right ring of quotients (Anh, 3 Oct 2025). This convergence is one of the clearest modern realizations of Jacobson localization as an operational technique rather than a purely formal slogan.

5. Sato modules, link modules, and factorization of free polynomials

If $I$10 is free of rank $I$11 as a right module with trivial left annihilator, a left $I$12-module $I$13 is a weak Sato module with respect to $I$14 if $I$15. Equivalently, the canonical map

$I$16

is an isomorphism. If in addition $I$17 is finitely presented, it is a Sato module (Anh, 3 Oct 2025). For such a module, the uniqueness of this decomposition produces endomorphisms $I$18 satisfying

$I$19

and these $I$20 define a left $I$21-module structure on $I$22 (Anh, 3 Oct 2025).

When $I$23 and $I$24, link modules in the sense of Sato and Farber–Vogel become Sato modules and hence natural modules over the Fox algebra $I$25 (Anh, 3 Oct 2025). A key structural statement is that Sato modules over skew free algebras or skew free group algebras are precisely the cokernels of matrices in $I$26 (Anh, 3 Oct 2025). This ties module theory directly to the Jacobson/Cohn localization condition that such matrices become invertible.

The main application is factorization theory for free polynomials with non-zero augmentation in the free group algebra. If $I$27 is comonic, meaning $I$28, then $I$29 is a Sato module (Anh, 3 Oct 2025). Theorem 7.10 states that $I$30 is irreducible in $I$31 if and only if $I$32 is simple as an $I$33-module if and only if the smallest lattice $I$34 is simple as a $I$35-module (Anh, 3 Oct 2025). Theorem 7.11 then identifies irreducible factorizations

$I$36

with composition chains in $I$37 (Anh, 3 Oct 2025).

The significance is precise: Jacobson localization supplies the ambient ring in which Fox derivatives act, Sato modules become linearized, and factorization is translated into module-theoretic composition structure. This suggests a broader methodological role for Jacobson localization in noncommutative factorization theory whenever an ideal topology and a distinguished generating row are available.

6. Jacobson-type properties in Ore extensions and profinite algebras

A broader ring-theoretic use of the phrase “Jacobson localization” concerns localizations or constructions controlled by the Jacobson radical and the Jacobson spectrum (Smoktunowicz et al., 2013). In this sense, recent results show that Jacobson behavior is highly sensitive to passage to differential polynomial rings. If $I$38 is the differential polynomial ring with multiplication determined by

$I$39

then there exists a locally nilpotent $I$40-algebra $I$41 and a derivation $I$42 such that $I$43 is not Jacobson radical (Smoktunowicz et al., 2013). At the same time, if $I$44 is an algebra over an uncountable field $I$45 and $I$46 is a locally nilpotent derivation, then

$I$47

is a nil ideal of $I$48 (Smoktunowicz et al., 2013).

These results delimit what Jacobson-controlled localization can and cannot infer from the coefficient ring. The negative example shows that local nilpotence of the base ring does not force Jacobson radicality of the differential Ore extension, so one cannot simply transfer radical behavior from $I$49 to $I$50 (Smoktunowicz et al., 2013). The positive intersection result shows, however, that under the uncountable-field and locally nilpotent-derivation hypotheses, the coefficient-level part of the Jacobson radical remains nil, which is exactly the kind of control used in localization arguments (Smoktunowicz et al., 2013).

An analogous tameness appears in profinite algebras. A profinite algebra is an inverse limit of finite dimensional algebras and can be written as $I$51 for a coalgebra $I$52 (Iovanov, 2015). With $I$53, Theorem 1.6 states that for a profinite algebra $I$54, the following are equivalent: $I$55 is left or right semiartinian; $I$56 is left or right perfect; $I$57 is semilocal and semiprimary, meaning the Jacobson radical is nilpotent; and $I$58 is semilocal and $I$59 is left or right T-nilpotent (Iovanov, 2015). Equivalently, in semilocal profinite algebras, T-nilpotence of $I$60 implies nilpotence (Iovanov, 2015).

For Jacobson localization, the profinite result means that in the semilocal/semiartinian pseudocompact setting, the Jacobson radical behaves as in a finite-length artinian situation: it is nilpotent, quotients $I$61 stabilize, and the Jacobson topology truncates after finitely many steps (Iovanov, 2015). In contrast, the differential-Ore counterexample shows that even very nil coefficient rings can develop non-quasi-regular elements after extension (Smoktunowicz et al., 2013). Taken together, these results indicate that Jacobson localization is effective when the surrounding radical structure is controlled by completion, finite Loewy length, or nilpotent filtration, but can fail to commute with noncommutative extension processes such as differential Ore extension.

7. Scope, limitations, and conceptual position

Across these works, Jacobson localization has a precise but nonuniform meaning. In the constructive sense of rational closure, it is the smallest subring of an $I$62-adically complete overring closed under Jacobson quasi-inverses, and in the free, free-group, and skew-free settings it coincides with a universal Cohn localization inverting matrices $I$63 over an augmentation-type ideal (Anh, 3 Oct 2025). In the broader structural sense, it refers to localization procedures sensitive to the Jacobson radical, maximal ideals, primitive ideals, or nilpotent radical layers (Smoktunowicz et al., 2013).

Its limitations are equally explicit in the literature. The work on differential polynomial rings does not explicitly carry out localization constructions and does not study $I$64 for multiplicative subsets $I$65 (Smoktunowicz et al., 2013). Rather, it identifies obstructions and controlled cases relevant to future localization theory. Conversely, the profinite-algebra results show that in semilocal profinite contexts, pathologies associated with T-nilpotent but non-nilpotent radicals do not occur, so Jacobson-localization-related behavior collapses to the semiprimary case (Iovanov, 2015).

A plausible synthesis is that Jacobson localization is best understood as a noncommutative localization paradigm driven by radical-theoretic data. In one direction, it produces explicit localizations through quasi-inverses, ideal topologies, and matrix inversion. In another, it functions as a diagnostic framework for asking whether radical behavior survives under extension, completion, duality, or passage to operator-like rings. The current literature shows both sides sharply: constructive realizability in Fox and skew-free settings, and nontrivial obstructions in differential Ore extensions.