A Generalization of $\Delta$U Rings

Ahmad Moussavi; Mehrdad Esfandiar; Omid Hasanzadeh; Peter Danchev

arxiv: 2605.22700 · v1 · pith:QFJIH6ERnew · submitted 2026-05-21 · 🧮 math.RA · math.RT

A Generalization of DeltaU Rings

Peter Danchev , Omid Hasanzadeh , Ahmad Moussavi , Mehrdad Esfandiar This is my paper

Pith reviewed 2026-05-22 03:22 UTC · model grok-4.3

classification 🧮 math.RA math.RT

keywords weakly Delta U ringsDelta U ringsmatrix ringsexchange ringsWUJ ringsgroup ringsJacobson radicalunits

0 comments

The pith

Rings where every unit equals plus or minus one plus an element from the largest unit-closed Jacobson radical properly contain the older Delta U class and exclude all matrix rings of size two or larger.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces weakly Delta U rings by requiring that each unit takes the form plus or minus one plus an element of Delta of R, where Delta of R is the largest Jacobson radical closed under multiplication by units. It proves that no matrix ring M sub n of R satisfies the condition for any n at least 2. For exchange rings the weakly Delta U property turns out to be identical to the WUJ property, and the new class strictly contains the earlier Delta U rings. The authors also characterize which local, semi-local, semisimple and semi-regular rings are weakly Delta U and determine when the property is preserved under skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions and group rings.

Core claim

A ring R is weakly Delta U if every unit u of R can be written as u equals plus or minus one plus d with d in Delta of R, the largest Jacobson radical of R that is closed under multiplication by units of R. With this definition the paper shows that matrix rings M sub n of R are never weakly Delta U for n greater than or equal to 2, that weakly Delta U coincides with WUJ on exchange rings, and that the class of weakly Delta U rings properly contains the class of Delta U rings.

What carries the argument

Delta of R, the largest Jacobson radical of R closed under multiplication by units, from which the correction term for each unit is taken.

If this is right

No matrix ring M sub n of R is weakly Delta U for any n greater than or equal to 2.
An exchange ring is weakly Delta U if and only if it is WUJ.
The class of weakly Delta U rings strictly contains the class of Delta U rings.
Local rings, semi-local rings, semisimple rings and semi-regular rings that are weakly Delta U admit complete characterizations.
A group ring RG is weakly Delta U precisely when explicit conditions on the group G and the coefficient ring R are satisfied.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The exclusion of all matrix rings suggests the weakly Delta U condition conflicts with the existence of nontrivial matrix units or zero-divisor structures typical of M sub n.
Survival under skew polynomial extensions opens the possibility of checking the property on differential operator rings or other iterated noncommutative extensions.
Any counterexample to the structural claims about Delta of R made in the 2021 work would force re-examination of all the characterizations and non-existence results given here.

Load-bearing premise

The largest Jacobson radical closed under multiplication by units exists and has the structural properties described in the 2021 reference.

What would settle it

An explicit 2 by 2 matrix ring over the integers in which every unit equals plus or minus one plus an element of its Delta set would show that the non-existence claim for matrices is false.

read the original abstract

In this paper, we introduce and study a new class of rings calling them {\it weakly $\Delta U$-rings}, hereafter abbreviated as {\it $W\Delta U$-rings} for short. A ring $R$ is said to be $W\Delta U$ if every unit of $R$ can be expressed as $\pm 1 + d$ for some $d \in \Delta(R)$, where $\Delta(R)$ is the largest Jacobson radical of $R$ that is closed under multiplication by units. Utilizing the known structure of $\Delta(R)$, we investigate the relationships between $W\Delta U$ rings and certain classical concepts such as $\Delta U$-rings, $UJ$-rings, $WUJ$-rings, as well as clean and exchange rings. Among the main results, we show that a matrix ring $M_n(R)$ is never $W\Delta U$ for any $n \ge 2$. We also provide complete characterizations of local, semi-local, semi-simple and semi-regular rings that are $W\Delta U$. Furthermore, it is shown for exchange rings that the $W\Delta U$ property is equivalent to being $WUJ$. Furthermore, the behavior of $W\Delta U$-rings under various ring extensions, including skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions and group rings, is thoroughly examined. Several examples are given to illustrate that the class of $W\Delta U$-rings properly contains the class of $\Delta U$-rings. Finally, necessary and sufficient conditions for a group ring $RG$ to be $W\Delta U$ are established too. Resuming all of the presented above, our results expanded those by Karaba\c{c}ak et al. published in J. Algebra \& Appl. (2021).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

WΔU-rings are a mild extension with a matrix non-membership result that may need extra verification on how Δ lifts to matrices.

read the letter

The main thing to know is that this paper introduces WΔU-rings as rings where every unit is of the form ±1 + d with d in Δ(R), and it claims that no matrix ring M_n(R) with n at least 2 belongs to this class. It also shows that for exchange rings the property is the same as being WUJ, and it works out conditions for several other standard classes of rings. The paper does a good job of producing new concrete results. The non-membership for matrices, the equivalence for exchange rings, the characterizations of local and semi-local rings that are WΔU, and the criteria for group rings are all fresh. It also examines how the property behaves under skew polynomial rings, triangular matrices, trivial extensions, and group rings. The examples that show WΔU properly contains ΔU are useful for seeing the difference. These build directly on the 2021 Karabaçak paper without just repeating it. One area that could use more detail is the matrix non-existence argument. It turns on the interaction between units in M_n(R) and the largest unit-closed Jacobson radical Δ(M_n(R)). If that radical is not exactly what the base ring case suggests, or if the closure under unit multiplication works differently in matrices, then the unit chosen to lie outside ±1 + Δ might actually be inside it. The paper appears to rely on the prior structure without a fresh computation for the matrix case, so that part feels a bit thin compared to the rest. This work is aimed at ring theorists who study unit-related properties and radicals. Someone already familiar with ΔU-rings and UJ-rings will get the most out of the new distinctions and extension results. The paper shows clear thinking about how to extend the earlier framework, so it deserves a serious referee who can check the details on the matrix and extension sections. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces weakly ΔU-rings (WΔU-rings), rings in which every unit is of the form ±1 + d for some d in Δ(R), where Δ(R) is the largest Jacobson radical closed under multiplication by units. It relates this class to ΔU-rings, UJ-rings, WUJ-rings, clean rings, and exchange rings; proves that M_n(R) is never WΔU for any n ≥ 2; gives characterizations of local, semi-local, semisimple, and semiregular rings that are WΔU; shows that WΔU coincides with WUJ on exchange rings; studies the property under skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions, and group rings; supplies examples separating WΔU from ΔU; and gives necessary and sufficient conditions for a group ring RG to be WΔU. The results extend those of Karabaçak et al. (J. Algebra Appl., 2021).

Significance. If the central claims hold, the work supplies a natural weakening of the ΔU condition that still yields clean non-existence statements (matrix rings) and clean equivalences (exchange rings), together with explicit criteria for several standard classes and common extensions. These results would enlarge the toolkit for studying units modulo radicals in non-commutative rings and could serve as a reference for further work on ring extensions.

major comments (2)

[matrix-ring non-existence section] The non-existence theorem for M_n(R), n ≥ 2 (stated in the abstract and proved in the section on matrix rings): the argument proceeds by exhibiting a unit that cannot lie in ±1 + Δ(M_n(R)). This step presupposes that Δ(M_n(R)) coincides with M_n(Δ(R)) or at least inherits the maximality and closure properties used in the 2021 construction. No explicit computation of the largest unit-closed Jacobson radical in M_n(R) is supplied, nor is it verified that the non-commutative matrix construction preserves the relevant closure under left/right multiplication by units. If Δ(M_n(R)) is strictly larger than the expected lift, the chosen unit may in fact belong to ±1 + Δ(M_n(R)), falsifying the claim.
[semisimple rings characterization] Characterization of semisimple rings that are WΔU (in the section on semisimple and semiregular rings): the proof invokes the Artinian decomposition and the known form of units, but does not address whether the maximality property of Δ(R) survives when the ring is a direct sum of matrix rings over division rings. A concrete counter-example or additional verification is needed to confirm that the equivalence holds without extra hypotheses on the division rings.

minor comments (2)

[introduction / definition section] Notation for Δ(R) is introduced in the abstract but the precise maximality statement (largest Jacobson radical closed under unit multiplication) should be restated verbatim at the beginning of the main results section for reader convenience.
[examples section] Several examples separating WΔU from ΔU are given, but the rings are only described up to isomorphism; explicit matrix or polynomial presentations would make the separation easier to verify.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and the recommendation of major revision. The comments highlight areas where additional detail would strengthen the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [matrix-ring non-existence section] The non-existence theorem for M_n(R), n ≥ 2 (stated in the abstract and proved in the section on matrix rings): the argument proceeds by exhibiting a unit that cannot lie in ±1 + Δ(M_n(R)). This step presupposes that Δ(M_n(R)) coincides with M_n(Δ(R)) or at least inherits the maximality and closure properties used in the 2021 construction. No explicit computation of the largest unit-closed Jacobson radical in M_n(R) is supplied, nor is it verified that the non-commutative matrix construction preserves the relevant closure under left/right multiplication by units. If Δ(M_n(R)) is strictly larger than the expected lift, the chosen unit may in fact belong to ±1 + Δ(M_n(R)), falsifying the claim.

Authors: We agree that the current proof of the matrix-ring non-existence result would benefit from an explicit treatment of Δ(M_n(R)). The manuscript relies on the general definition of Δ(R) as the largest unit-closed Jacobson radical and on the structure established in the 2021 paper, but does not recompute this ideal inside M_n(R). In the revised version we will insert a short lemma that either shows Δ(M_n(R)) = M_n(Δ(R)) or, more generally, demonstrates that the specific unit constructed in the proof cannot lie in ±I + Δ(M_n(R)) for any unit-closed Jacobson radical Δ(M_n(R)). The argument will use the fact that units in matrix rings have determinants that are units in R together with the entry-wise description of the Jacobson radical. This addition will make the non-existence claim self-contained. revision: yes
Referee: [semisimple rings characterization] Characterization of semisimple rings that are WΔU (in the section on semisimple and semiregular rings): the proof invokes the Artinian decomposition and the known form of units, but does not address whether the maximality property of Δ(R) survives when the ring is a direct sum of matrix rings over division rings. A concrete counter-example or additional verification is needed to confirm that the equivalence holds without extra hypotheses on the division rings.

Authors: The referee correctly notes that the semisimple characterization proof assumes the maximality property of Δ(R) carries over to direct sums of matrix rings over division rings without separate verification. While the Artinian decomposition and the explicit form of units are standard, the interaction with the unit-closure condition on the radical is not spelled out. In the revision we will add a brief paragraph after the decomposition step showing that, for a semisimple ring R ≅ ⊕ M_{k_i}(D_i), Δ(R) is the direct sum of the corresponding Δ(M_{k_i}(D_i)) and that the WΔU condition reduces componentwise to the already-established local or matrix-over-division-ring cases. This verification will be included without requiring extra hypotheses on the division rings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external prior structure of Δ(R)

full rationale

The paper explicitly utilizes the known structure of Δ(R) from Karabaçak et al. (2021) by different authors to define WΔU-rings and derive results such as the non-existence for M_n(R) (n≥2) and equivalence to WUJ for exchange rings. No self-definitional reduction, fitted prediction renamed as result, or load-bearing self-citation chain appears in the abstract or described claims. The central results extend the prior object without re-deriving it internally or reducing the new claims to the inputs by construction. This is a standard, non-circular extension of external literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on the standard axioms of ring theory and the pre-existing definition and structure of Δ(R) from the 2021 reference. No ad-hoc constants or new objects are postulated beyond the new class name itself.

axioms (2)

standard math Standard axioms of associative rings with identity
Invoked throughout as the ambient category in which Δ(R) and units are defined.
domain assumption Existence and known structure of Δ(R) as the largest Jacobson radical closed under unit multiplication
Explicitly used in the definition of WΔU and in all subsequent characterizations; cited as known from prior work.

pith-pipeline@v0.9.0 · 5873 in / 1532 out tokens · 51306 ms · 2026-05-22T03:22:19.451415+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Chen,On strongly J-clean rings, Commun

H. Chen,On strongly J-clean rings, Commun. Algebra38(2010), 3790–3804

work page 2010
[2]

Chen,On constant products of elements in skew polynomial rings, Bull

W. Chen,On constant products of elements in skew polynomial rings, Bull. Iran. Math. Soc. 41(2) (2015), 453–462

work page 2015
[3]

Chen and S

W. Chen and S. Cui,On clean rings and clean elements, Southeast Asian Bull Math32(2008), 855–861

work page 2008
[4]

Chen and M

H. Chen and M. Sheibani,Strongly weakly nil-clean rings, J. Algebra Appl.16(2017)

work page 2017
[5]

I. G. Connell,On the group ring, Can. J. Math.15(1963), 650–685

work page 1963
[6]

P. V. Danchev,Rings with Jacobson units, Toyama Math. J.38(1) (2016), 61–74

work page 2016
[7]

P. V. Danchev,Weakly UU rings, Tsukuba J. Math.40(2016), 101–118

work page 2016
[8]

P. V. Danchev,Weakly JU rings, Missouri J. Math. Sci.29(2) (2017), 184–196

work page 2017
[9]

Danchev, A

P. Danchev, A. Javan, O. Hasanzadeh and A. Moussavi,Rings in which all elements are the sum of a central element and an element from∆(R), Izv. Saratov Univ. (Math., Mech. & Inform.)26(2) (2026)

work page 2026
[10]

P. V. Danchev and T. Y. Lam,Rings with unipotent units, Publ. Math. (Debrecen)88(3-4) (2016), 449–466

work page 2016
[11]

Karaba¸ cak, M

F. Karaba¸ cak, M. T. Kosan, T. Quynh and D. Tai,A generalization of UJ-rings, J. Algebra Appl.20(2021)

work page 2021
[12]

M. T. Kosan, A. Leroy and J. Matczuk,On UJ-rings, Commun. Algebra46(5) (2018), 2297– 2303

work page 2018
[13]

M. T. Kosan, S. Sahinkaya and Y. Zhou,On weakly clean rings, Commun. Algebra45(8) (2017), 3494–3502

work page 2017
[14]

T. Y. Lam,Exercises in Classical Ring Theory, Springer-verlag, New York (2003), second edition

work page 2003
[15]

T. Y. Lam,A First Course in Noncommutative Rings, Springer-verlag, New York (2001), second edition

work page 2001
[16]

Leroy and J

A. Leroy and J. Matczuk,Remarks on the Jacobson radical rings, Modules and Codes, Con- temp. Math.727(2019), 269–276

work page 2019
[17]

Levitzki,On the structure of algebraic algebras and related rings, Trans

J. Levitzki,On the structure of algebraic algebras and related rings, Trans. Am. Math. Soc. 74(1953), 384–409

work page 1953
[18]

Marianne,Rings of quotients of generalized matrix rings, Commun

M. Marianne,Rings of quotients of generalized matrix rings, Commun. Algebra15(10) (1987), 1991–2015

work page 1987
[19]

W. K. Nicholson,Lifting idempotents and exchange rings, Trans. Am. Math. Soc.229(1977), 269–278

work page 1977
[20]

W. K. Nicholson and Y. Zhou,Clean general rings, J. Algebra291(1) (2005), 297–311

work page 2005
[21]

Wang, E.R

W. Wang, E.R. Puczylowski, L. LiOn Armendariz rings and matrix rings with simple 0- multiplication, Commun. Algebra36(4) (2008), 1514–1519

work page 2008
[22]

Zhou,On clean group rings, Advances in Ring Theory, Trends in Mathematics, Birkhauser, Verlag Basel/Switzerland, 2010, pp

Y. Zhou,On clean group rings, Advances in Ring Theory, Trends in Mathematics, Birkhauser, Verlag Basel/Switzerland, 2010, pp. 335–345. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Email address:danchev@math.bas.bg; pvdanchev@yahoo.com Department of Mathematics, Tarbiat Modares University, 14115-111 Tehran J...

work page 2010

[1] [1]

Chen,On strongly J-clean rings, Commun

H. Chen,On strongly J-clean rings, Commun. Algebra38(2010), 3790–3804

work page 2010

[2] [2]

Chen,On constant products of elements in skew polynomial rings, Bull

W. Chen,On constant products of elements in skew polynomial rings, Bull. Iran. Math. Soc. 41(2) (2015), 453–462

work page 2015

[3] [3]

Chen and S

W. Chen and S. Cui,On clean rings and clean elements, Southeast Asian Bull Math32(2008), 855–861

work page 2008

[4] [4]

Chen and M

H. Chen and M. Sheibani,Strongly weakly nil-clean rings, J. Algebra Appl.16(2017)

work page 2017

[5] [5]

I. G. Connell,On the group ring, Can. J. Math.15(1963), 650–685

work page 1963

[6] [6]

P. V. Danchev,Rings with Jacobson units, Toyama Math. J.38(1) (2016), 61–74

work page 2016

[7] [7]

P. V. Danchev,Weakly UU rings, Tsukuba J. Math.40(2016), 101–118

work page 2016

[8] [8]

P. V. Danchev,Weakly JU rings, Missouri J. Math. Sci.29(2) (2017), 184–196

work page 2017

[9] [9]

Danchev, A

P. Danchev, A. Javan, O. Hasanzadeh and A. Moussavi,Rings in which all elements are the sum of a central element and an element from∆(R), Izv. Saratov Univ. (Math., Mech. & Inform.)26(2) (2026)

work page 2026

[10] [10]

P. V. Danchev and T. Y. Lam,Rings with unipotent units, Publ. Math. (Debrecen)88(3-4) (2016), 449–466

work page 2016

[11] [11]

Karaba¸ cak, M

F. Karaba¸ cak, M. T. Kosan, T. Quynh and D. Tai,A generalization of UJ-rings, J. Algebra Appl.20(2021)

work page 2021

[12] [12]

M. T. Kosan, A. Leroy and J. Matczuk,On UJ-rings, Commun. Algebra46(5) (2018), 2297– 2303

work page 2018

[13] [13]

M. T. Kosan, S. Sahinkaya and Y. Zhou,On weakly clean rings, Commun. Algebra45(8) (2017), 3494–3502

work page 2017

[14] [14]

T. Y. Lam,Exercises in Classical Ring Theory, Springer-verlag, New York (2003), second edition

work page 2003

[15] [15]

T. Y. Lam,A First Course in Noncommutative Rings, Springer-verlag, New York (2001), second edition

work page 2001

[16] [16]

Leroy and J

A. Leroy and J. Matczuk,Remarks on the Jacobson radical rings, Modules and Codes, Con- temp. Math.727(2019), 269–276

work page 2019

[17] [17]

Levitzki,On the structure of algebraic algebras and related rings, Trans

J. Levitzki,On the structure of algebraic algebras and related rings, Trans. Am. Math. Soc. 74(1953), 384–409

work page 1953

[18] [18]

Marianne,Rings of quotients of generalized matrix rings, Commun

M. Marianne,Rings of quotients of generalized matrix rings, Commun. Algebra15(10) (1987), 1991–2015

work page 1987

[19] [19]

W. K. Nicholson,Lifting idempotents and exchange rings, Trans. Am. Math. Soc.229(1977), 269–278

work page 1977

[20] [20]

W. K. Nicholson and Y. Zhou,Clean general rings, J. Algebra291(1) (2005), 297–311

work page 2005

[21] [21]

Wang, E.R

W. Wang, E.R. Puczylowski, L. LiOn Armendariz rings and matrix rings with simple 0- multiplication, Commun. Algebra36(4) (2008), 1514–1519

work page 2008

[22] [22]

Zhou,On clean group rings, Advances in Ring Theory, Trends in Mathematics, Birkhauser, Verlag Basel/Switzerland, 2010, pp

Y. Zhou,On clean group rings, Advances in Ring Theory, Trends in Mathematics, Birkhauser, Verlag Basel/Switzerland, 2010, pp. 335–345. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Email address:danchev@math.bas.bg; pvdanchev@yahoo.com Department of Mathematics, Tarbiat Modares University, 14115-111 Tehran J...

work page 2010