Nilpotency Index Research Articles

Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models.

О перманенте многомерных матриц

The permanent of multidimensional matrices is expressed in terms of operations on elements of commutative algebra with nilpotent index 2 generators. Using a technique based on this relationship, several properties of the permanent have been proved. Various types of multidimensional permutations are considered. The permanent of multidimensional matrices is considered from the point of view of the enumeration function of multidimensional permutations.

One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has aradical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremumbeing finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J3 = 0 and without considering that rad2 (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case.

The radical of functorially finite subcategories

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if radC∞ vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.

Tensor product of finite dimensional nilpotent evolution algebras

This paper investigates the tensor product of a finite dimensional nilpotent evolution algebra. Some properties that translate from tensor products to factors and vice versa have been investigated, including the index of nilpotency and annihilator. The index of nilpotency of the tensor product of two nilpotent evolution algebras with different indexes of nilpotency is determined. Moreover, we investigate the tensorially decomposable of the 4-dimensional nilpotent evolution algebra. In addition, the decomposable nilpotent evolution algebra with the maximal index of nilpotency has been carried out.

Construction of Rank-One Solvable Rigid Lie Algebras with Nilradicals of a Decreasing Nilpotence Index

It is shown that for any integers k≥2, q≥2k and N≥k+q+2, there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations t possessing the eigenvalue spectrum spec(t)=1,2,⋯,k,q,q+1⋯,N, a nilradical of the nilpotence index N−k and a characteristic sequence (N−k,1k).

When nilpotent elements generate nilpotent ideals

We study the natural class of rings where each nilpotent element generates a nilpotent ideal, calling them the strongly 2-primal rings. We derive many basic properties of these rings, analyze their behavior under standard ring constructions and extensions, and taxonomize their relationship to other natural generalizations of commutativity. A slightly stronger condition is to assume that any nilpotent element generates a nilpotent ideal of the same index of nilpotence. We find that the difference between these two properties explains divergent behaviors in direct products, polynomial rings, Morita contexts, and other constructions.

Third power associative rings satisfying a(bc)=b(ca)

In this paper, we study third power associative rings satisfying the identity We prove that third power associative rings satisfying the identity with characteristics are power associative and associators generate nilpotent ideal with index of nilpotency . Furthermore, we prove that third power associative rings satisfying the identity with commutators contained in the left nucleus and characteristics are associative and commutative of degree five.

Left invariant Ricci flat metrics on Lie groups

Abstract In this paper, we apply the double extension process to study left invariant Ricci flat metrics on solvable and non-solvable Lie groups. An inductive method to produce new Ricci flat metrics from the old ones is established. As applications, we prove the following two results: (i) Every nilpotent Lie group with dim ⁡ C ⁢ ( G ) ≥ 1 2 ⁢ ( dim ⁡ G - 1 ) {\dim\mathrm{C}(G)\geq\frac{1}{2}(\dim G-1)} admits a left invariant Ricci flat metric. (ii) Given a Lie group G, there exists a nilpotent Lie group N with nilpotent index at most 2 such that G × N {G\times N} admits a left invariant Ricci flat metric. We also construct infinitely many new explicit examples of left invariant Ricci flat metrics on nilpotent Lie groups.

On identities of model algebras

A sharp upper bound for the nilpotency index of the commutator ideal of a $2$-generated subalgebra of an arbitrary model algebra is given; this estimate is about half that for arbitrary Lie nilpotent algebras of the same class. All identities in two variables that hold in the model algebra of multiplicity $3$ are found. For any $m\geqslant 3$, in a free Lie nilpotent algebra $F^{(2m+1)}$ of class $2m$, the kernel polynomial of smallest possible degree is indicated. It is proved that the degree of any identity of a model algebra is greater than its multiplicity.

On the nilpotency index of the radical of a module category

Let A be a finite dimensional representation-finite algebra over an algebraically closed field. The aim of this work is to determine which vertices of QA are sufficient to be considered in order to compute the nilpotency index of the radical of the module category of A. Precisely, we study the above mentioned problem whenever A is a monomial or a toupie algebra where their Auslander-Reiten quivers are component with length. We also study the above problem for string algebras. We give concrete formulas to compute such index for many classes of algebras such as toupie algebras, tree algebras and some classes of pullback algebras over the hereditary algebras of type An, Dn and E6. Furthermore, we determine the index taking into account the ordinary quiver of the given algebras.

On properties of some nilpotent evolution algebras and their enveloping algebras

In this paper, we consider [Formula: see text]-dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra.

Quotient algebras of Banach operator ideals related to non-classical approximation properties

We investigate the quotient algebra AXI:=I(X)/F(X)‾||⋅||I for Banach operator ideals I contained in the ideal of the compact operators, where X is a Banach space that fails the I-approximation property. The main results concern the nilpotent quotient algebras AXQNp and AXSKp for the quasi p-nuclear operators QNp and the Sinha-Karn p-compact operators SKp. The results include the following: (i) if X has cotype 2, then AXQNp={0} for every p≥1; (ii) if X⁎ has cotype 2, then AXSKp={0} for every p≥1; (iii) the exact upper bound of the index of nilpotency of AXQNp and AXSKp for p≠2 is max⁡{2,⌈p/2⌉}, where ⌈p/2⌉ denotes the smallest n∈N such that n≥p/2; (iv) for every p>2 there is a closed subspace X⊂c0 such that both AXQNp and AXSKp contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace X⊂c0 such that the compact-by-approximable algebra AX=K(X)/A(X) contains two incomparable countably infinite chains of nilpotent closed ideals.

A CHARACTERISATION OF MATRIX RINGS

AbstractWe prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$ . As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs [‘A characterisation result for matrix rings’, Bull. Aust. Math. Soc.43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R, we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so.

Symmetric Zinbiel superalgebras

The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.

Functional degrees and arithmetic applications I: The set of functional degrees

We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any finitely generated commutative groups A and B, we compute the complete set D(A,B) of functional degrees of all maps between A and B. In particular, we see how big the functional degree of a function with finite functional degree can get, in which cases there is a maximal finite functional degree, and when there are functions of infinite functional degree between A and B. This yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form Zpβ[Zpα1×Zpα2×⋯×Zpαn] with p,β,n,α1,…,αn∈Z+, p prime.

Superalgebras: Polynomial identities and asymptotics

To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞⁡cnsup(A)n is an integer that can be explicitly computed.Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then(1)C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group.As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that limn→∞⁡logn⁡cnsup(A)expsup(A)n exists and is a half integer.

On the enumeration and classification of σ-LCD codes over finite commutative chain rings

Let Re denote a finite commutative chain ring with the maximal ideal 〈u〉 and the nilpotency index e, and let R‾e=Re/〈u〉 be the residue field of Re. Let σ0 be an automorphism of Re, and let σ‾0 be the corresponding automorphism of the residue field R‾e of Re, defined as σ‾0(a+〈u〉)=σ0(a)+〈u〉 for all a+〈u〉∈R‾e. Let σ be an automorphism of Ren corresponding to the automorphism σ0 of Re, defined as σ(v1,v2,…,vn)=(σ0(v1),σ0(v2),…,σ0(vn)) for all (v1,v2,…,vn)∈Ren. In this paper, we obtain explicit enumeration formulae for all σ-LCD codes of an arbitrary length over the chain ring Re when σ‾02 is the identity automorphism of R‾e. With the help of these enumeration formulae and by applying the classification algorithm, we classify all Euclidean LCD codes of lengths 2, 3, 4 and 5 over the chain ring F2[u]/〈u2〉 and of lengths 2, 3 and 4 over the chain ring F3[u]/〈u2〉, and all σ-LCD codes of lengths 2, 3 and 4 over the chain ring F4[u]/〈u2〉, where σ0 is an automorphism of F4[u]/〈u2〉 such that the corresponding automorphism σ‾0 of the residue field F4 has order 2. Besides this, we show that the class of σ-LCD codes over Re is asymptotically good, and that every free linear [n,k,d]-code over Re is equivalent to a σ-LCD [n,k,d]-code over Re when |R‾e|>4. We also explicitly determine all inequivalent σ-LCD [n,1,d]-codes and [n,n−1,d]-codes over Re for 1≤d≤n.

Dealing with the Resolubility of Evolution Algebras

Although since their introduction by Tian in 2004, evolution algebras have been the subject of a very deep study in the last years due to their numerous applications to other disciplines, this study is not easy since these algebras lack an identity that characterizes them, such as the identity of Jacobi, for Lie algebras or those of Leibniz and Malcev for those corresponding algebras. In this paper we deal with the concepts of solvability and nilpotency of these evolution algebras. Some novel results on them obtained from using the evolution operator of these algebras are given and some examples illustrating these results are also shown. The main result obtained states that an evolution algebra is solvable if and only if its structure matrix is nilpotent, which implies, in turn, that the solvability and the nilpotency indices of that algebra coincide provided the corresponding evolution operator is an endomorphism of the algebra.

On the radical of cluster tilted algebras

We determine the minimal lower bound n, with \(n \geqslant 1\), where the n-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of m, \(m \geqslant 2\), irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the \((m\,{+}\,1)\)-th power of the radical of their module category.