Nilpotency Index Research Articles

Linearly triangularizable quadratic endomorphisms

In the article, we prove that any polynomial endomorphism of , where is a field of characteristic zero, of the form ‘identity + homogeneous component of degree two’ is linearly triangularizable when the Jacobian matrix of the quadratic term is nilpotent of index at most three. In particular, this proves the well-known Meisters and Olech Conjecture. In our approach, besides methods of the classical matrix theory, some elements of the theory of commutative non-associative algebras appear in the context of the so-called polarization algebras associated with quadratic homogeneous endomorphisms. A powerful role of polarization algebras in the investigation of polynomial endomorphisms is revealed in a recent paper of Umirbaev. His results and comments were the main inspiration for us.

Generalized stretched ideals and Sally's conjecture

We introduce the concept of j-stretched ideal in a Noetherian local ring. It generalizes to arbitrary ideals the classical notion of stretched m-primary ideal of Sally and Rossi–Valla, as well as the concept of ideal of minimal and almost minimal j-multiplicity introduced by Polini and Xie. One of our main theorems states that, for a j-stretched ideal, the associated graded ring is Cohen–Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem gives numerical conditions ensuring the almost Cohen–Macaulayness of the associated graded ring of a j-stretched ideal, and it provides a generalized version of Sally's conjecture. This work, which also holds for modules, unifies the approaches of Rossi–Valla and Polini–Xie and generalizes simultaneously results on the Cohen–Macaulayness or almost Cohen–Macaulayness of the associated graded module by several authors, including Sally, Rossi–Valla, Wang, Elias, Corso–Polini–Vaz Pinto, Huckaba, Marley and Polini–Xie.

Special properties of a skew triangular matrix ring with constant diagonal

In this paper, we study various annihilator properties in a ring [Formula: see text] with an endomorphism [Formula: see text] and some subrings of the skew triangular matrix rings [Formula: see text]. They allow the construction of rings with a nonzero nilpotent ideal of arbitrary index of nilpotency which have various zero-divisor properties.

On Lie Nilpotent Rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z n (R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring ⟨Z n (R) ∪ C⟩ of R generated by the subset Z n (R) ∪ C of R is also Lie nilpotent of index n.

Metabelian Associative Algebras

Metabelian algebras are introduced and it is shown that an algebra $A$ is metabelian if and only if $A$ is a nilpotent algebra having the index of nilpotency at most $3$, i.e. $x y z t = 0$, for all $x$, $y$, $z$, $t \in A$. We prove that the It\^{o}'s theorem for groups remains valid for associative algebras. A structure theorem for metabelian algebras is given in terms of pure linear algebra tools and their classification from the view point of the extension problem is proven. Two border-line cases are worked out in detail: all metabelian algebras having the derived algebra of dimension $1$ (resp. codimension $1$) are explicitly described and classified. The algebras of the first family are parameterized by bilinear forms and classified by their homothetic relation. The algebras of the second family are parameterized by the set of all matrices $(X, Y, u) \in {\rm M}_{n}(k)^2 \times k^n$ satisfying $X^2 = Y^2 = 0$, $XY = YX$ and $Xu = Yu$.

Group algebras with almost minimal Lie nilpotency index

Group algebras with almost minimal Lie nilpotency index

A swiss cheese theorem for linear operators with two invariant subspaces

We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\to V$ and two $T$-invariant subspaces $U_1\subset U_2\subset V$. Let $\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $\mathcal S(n)$ for $n\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\leq 4$. But not each dimension type in the cylinder can be realized by an indecomposable system. In particular, there are holes in the cylinder. Namely, no triple in $(x,y,z)\in (3,1,3)+\mathbb N(2,2,2)$ can be realized, while each neighbor $(x\pm1,y,z), (x,y\pm1,z),(x,y,z\pm1)$ can. Compare this with Bongartz' No-Gap Theorem, which states that for an associative algebra $A$ over an algebraically closed field, there is no gap in the lengths of the indecomposable $A$-modules of finite dimension.

On Skew Triangular Matrix Rings

Let R be a ring with an endomorphism σ. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S(R,n,σ) and T(R,n,σ). They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which inherit various interesting properties of rings.

Derivations and bounded nilpotence index

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

Nilpotent Rings of Order p 4 and Nilpotency Index 3

We give a complete description of irreducible rings of order p 4 and nilpotency index 3 provided that the identity x 2 = 0 and partially describe nilpotent such rings if the identity x 2 = 0. fails.

Noncommutativity makes determinants hard

We consider the complexity of computing the determinant over arbitrary finite-dimensional algebras. We first consider the case that A is fixed. In this case, we obtain the following dichotomy: If A/radA is noncommutative, then computing the determinant over A is hard. “Hard” here means #P-hard over fields of characteristic 0 and ModpP-hard over fields of characteristic p>0. If A/radA is commutative and the underlying field is perfect, then we can compute the determinant over A in polynomial time.We also consider the case when A is part of the input. Here the hardness is closely related to the nilpotency index of the commutator ideal of A.Our work generalizes and builds upon previous papers by Arvind and Srinivasan (STOC 2010) [1] as well as Chien et al. (STOC 2011) [5].

Semiprime FDI-rings

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

On Some Zero-Filiform Algebras

We present the description of algebras with the maximum nilpotency index given by certain special identities.

Commutator algebras arising from splicing operations

Abstract We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and we formulate a series of open problems.

A new class of non-semiprime quasi-Armendariz rings

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

Graded cocharacters of minimal subvarieties of supervarieties of almost polynomial growth

The sequence of graded codimensions of a minimal supervariety of polynomial growth grows like nt, but any proper subvariety grows like ns with s<t. Such supervarieties are the building blocks of general supervarieties of polynomial growth. In this paper, we exhibit the decompositions of the graded cocharacters of all minimal subvarieties of supervarieties of almost polynomial growth and compute their graded colengths. Furthermore, for a finite dimensional superalgebra A such that its sequence of graded codimensions grows like nt we exhibit an upper bound for the nth graded colength of A which depends only on t and on two parameters related to A such as its dimension and the index of nilpotence of its Jacobson radical J(A).

From submodule categories to preprojective algebras

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\). Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\); the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\). Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\), thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.

Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus

Let f be a triangular automorphism of the affine N-space of degree d with Jacobian 1 over a \({\mathbb{Q}}\)-algebra R. We introduce a weighted nilpotency index ν(f) for f, and give a bound of deg(f n ) in terms of N, d and ν(f) for all \({n \in \mathbb{Z}}\). When N = 2, our formula, combined with computation of the Hilbert series of certain graded algebras, yields the estimate deg(f n ) ≤ d 2 − d + 1 for all \({n \in \mathbb{Z}}\). If n varies through all integers, this estimate turns out to be sharp and is related, somewhat unexpectedly, to the Schubert calculus on the Grassmannian G(d − 1, 2d − 2). Numerical computation for small degrees suggests that this estimate restricted to the inverse degree (i.e. n = −1) is also sharp if d ≥ 3.

A note on nilpotent rings

Let R be a ring on an Abelian group A. The symbol Rn denotes the set of all finite sums of products of n elements of R. The smallest positive integer n such that Rn = 0 is called the index of nilpotency of R and is denoted by I(R) = n. In this paper, we study some nilpotent rings. In particular, we determine the index of nilpotency of some torsion-free rings of finite rank.

On the relationship between the length of an algebra and the index of nilpotency of its Jacobson radical

A sharp upper bound for the length of an algebra as a function of the index of nilpotency of its Jacobson radical and the length of the quotient algebra by the radical is obtained.