Agata Smoktunowicz, Laurent Bartholdi
For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl.
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It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil rings, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some …
We answer a question by Shestakov on the Jacobson radical in differential polynomial rings. We show that if R is a locally nilpotent ring with a derivation D then R[X;D] need not be Jacobson radical. We also show that J(R[X;D])\cap R is a nil ideal of R in the case where D is a locally nilpotent derivation and R is an algebra over an uncountable field.
Given a t.u.p.-semigroup (two unique product semigroup) $X$, we show if $R$ is an $X$-graded ring then both its nilradical and Jacobson radical are homogeneous. This partially answers questions of Smoktunowicz and Jespers.
We give a method to study the finiteness of the coradical filtration of a coalgebra; as a consequence, we show that a left semiartinian profinite algebra has nilpotent Jacobson radical and is right semiartinian too. Equivalently, we show that a for a semilocal profinite algebra, T-nilpotence implies nilpotence for the Jacobson radical. This answers two open questions from \cite{INT}.
The quantum nilpotent algebras U^w_-(g), defined by De Concini-Kac-Procesi and Lusztig, are large classes of iterated skew polynomial rings with rich ring theoretic structure. In this paper, we prove in an explicit way that all torus invariant prime ideals of the algebras U^w_-(g) are polynormal. In the special case of the algebras of quantum matrices, this construction yields explicit polynormal generating sets consisting of quantum minors for all of their torus invariant prime ideals. This gives a constructive proof of the Goodearl-Lenagan polynormality conjecture. Furthermore we prove that …
Let $R$ be a ring satisfying a polynomial identity and let $D$ be a derivation of $R$. We consider the Jacobson radical of the skew polynomial ring $R[x;D]$ with coefficients in $R$ and with respect to $D$, and show that $J(R[x;D])\cap R$ is a nil $D$-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when $R$ is commutative.
Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.
A well-known theorem by S.A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x; D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x; D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p>0.
A ring is said to satisfy the $2$-nil-sum property if every non central-unit is the sum of two nilpotents. We prove that a ring satisfies the $2$-nil-sum property iff it is either a simple ring with the $2$-nil-sum property or a commutative local ring with nil Jacobson radical, and we provide an example of a simple ring with the $2$-nil-sum property that is not commutative. Moreover, a simple right Goldie ring has the $2$-nil-sum property iff it is a field.
We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated: We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated, and on the other hand construct a finitely generated, infinite dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.
On differential polynomial rings with locally nilpotent derivations
该研究针对带局部幂零导子的微分多项式环的根结构问题，证明了其Jacobson根可表示为底环某诣零理想对应的微分多项式环，解决了Agata Smoktunowicz提出的相关问题。
For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$, the Jacobson--Morozov theorem gives a construction of subalgebras $\mathfrak{sl}_2 \subset \mathfrak{g}$ corresponding to nilpotent elements of $\mathfrak{g}$. In this note, we propose an extension of the Jacobson--Morozov theorem to the symmetrizable Kac--Moody setting and give a proof of this generalization in the case of rank two hyperbolic Kac--Moody algebras. We also give a proof for an arbitrary symmetrizable Kac--Moody algebra under some stronger restrictions.
We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial p_x with integer coefficients, such that p_x(1)=1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the Köthe conjecture, namely the integral rings.
On a $q$-Skew Amitsur's Theorem
本文研究q-斜Ore扩张的Jacobson radical，证明其常数部分为 nil，部分回答Greenfeld等2019年问题，并在特征为零时运用Shin结果证明q-斜阿米修尔定理。
Jacobson radicals of Ore extensions
本文研究Ore扩张$R[x;\sigma,D]$的Jacobson根，给出特征零下$q$-斜导子情形的结构公式，并证明在PI环或正特征局部幂零导子等条件下其与系数环的交为诣零理想。
Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These …
We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In §\ref{Section6}, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension $\geq 8$ over a countable field which answers a question from \cite{8}.
Some results on null ideals of finite rings
本文针对有限环的零理想展开研究，证明当环的Jacobson根幂零指数不超过3时其零理想为双侧理想，还构造出幂零指数为任意不小于5的整数的反例环。
Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{σ(i_1)} x_{σ(i_2)} \cdots x_{σ(i_l)}$, where $σ$ runs through $H$, is considered. It is shown that $G$ has a free subgroup of finite index. For a field $K$, properties of the algebra $K[G]$ are derived. In particular, the Jacobson radical $\mathcal{J}(K[G])$ is always nilpotent, and in many cases the algebra $K[G]$ is semiprimitive. Results on the growth and the Gelfand-Ki…
Unit Uniquely Clean Rings
本文提出单位唯一清洁环（UnitUC），统一推广唯一清洁环与强诣零清洁环，并刻画其与布尔环、交换环及 Jacobson 根的关系，同时研究其在群环和矩阵环扩张下的表现。
The main result of this paper is the following: if F is any field and R is any F-subalgebra of the algebra of nxn matrices over F with Lie nilpotence index m, then the F-dimension of R is less or equal than M(m+1,n), where M(m+1,n) is the maximum of a certain expression of m+1 nonnegative integers and n. The case m=1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero, and R is any commutative F-subalgebra of the full nxn matrix algebra over F, then the F-dimension of…
Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are closely connected. The aim of the paper is to show that this is also true when B(R, G) is not empty set under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (resp., the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (resp.…
We prove that if A is a finite dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of …
It is shown that an algebra $Λ$ can be lifted with nilpotent Jacobson radical $r = r(Λ)$ and has a generalized matrix unit $\{e_{ii}\}_I$ with each $\bar e_{ii} $ in the center of $\bar Λ= Λ/r$ iff $Λ$ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, $Λ$ is a finite algebra with non-zero unity element over perfect field $k$ (e.g. a field with characteristic zero or a finite field) iff $Λ$ is isomorphic to a generalized path algebra $k (D, Ω, ρ)$ of finite directed graph with weak relations and $dim {\} …
We study solvable Lie algebras in prime characteristic $p$ that admit non-singular derivations. We show that Jacobson's Theorem remains true if the quotients of the derived series have dimension less than~$p$. We also study the structure of Lie algebras with non-singular derivations in which the derived subalgebra is abelian and has codimension~one. The paper presents some new examples of solvable, but not nilpotent, Lie algebras of derived length~3 with non-singular derivations.
Rings Whose Non-Units are Square-Nil Clean
本文研究强NUS-幂零清洁环（非单位元可表为交换的幂零元与平方幂等元之和），给出其判定准则，刻画仅含平凡幂等元时的等价条件，并提供矩阵构造与群环扩张的新例。
Rings Whose Non-Invertible Elements are Strongly Weakly Nil-Clean
本文研究非单位元素均为强弱幂零清洁的环，证明其为强π-正则，并给出基于Jacobson根和2×2矩阵环截断的完整刻画，同时探讨Morita上下文环和群环等扩展。
Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the spectrum Spec R of all prime ideals of R, viewed as a topological space with the Jacobson-Zariski topology, and on the subspace consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid of R/P is equal to the base field. The main themes of the article are local closedness of G-orbits in Spec R and the so-called G-stratification of Spec R. This stratification plays a central role…
We give an explicit description of adjoint quotient maps for Jacobson-Witt algebra Wn and special algebra Sn. An analogue of Kostant's differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of adjoint quotient map for Sn and construct the analogs of Kostant's transverse slice. In addition, we analysis the nilpotent elements and generalize some Premet's results to Sn.
Classical ideals theory of maximal subrings in non-commutative rings
本文研究非交换环中极大子环扩张下若干重要理想（如根、基座、奇异理想）的关系，给出特定条件下（如约化环、左阿廷环）的精确刻画与结构分解。
We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either…
In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Lin. & Multilin. Algebra (2021) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Lin. Algebra & Appl. (2018) and Shitov in Indag. Math. (2019).
Extremal Derived Length in the Unipotent Radicals of Cartan-Type Automorphism Groups
本文研究Cartan型自同构群幺幂根的导长度，证明特殊、Witt和Hamiltonian三类根均达到其幂零类允许的最大导长度，并给出相关滤过与序列的显式刻画。
Associative/Lie radical (co)invariance under tracial (co)actions
该研究针对霍普夫代数上的有限维结合或李（共）模代数，在特征合适且（共）作用保迹的条件下，证明了多种根基的（共）不变性结果，推广了已有准则并提供统一框架。
On Rings with the 2-UNJ Property
本文引入了推广2-UJ、2-UU和UNJ环的2-UNJ环，证明了包含关系并给出反例，同时探讨了其与多种代数性质的联系及自然扩张下的新刻画，为后续研究奠定基础。
Hopkins-Levitzki Type Theorems for Groupoid Graded Rings
该研究针对群胚分次环，引入分次理想的左/右逐对象幂零性概念，推广Hopkins-Levitzki定理，证明三类Γ₀-Artinian环为Γ₀-Noetherian环，并用反例说明相关结论。
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