Papers

In this paper, we utilize the reflexive hull to unify the concepts of local and almost inner derivations. By investigating the algebraic structure of reflexive hulls, we prove the conjecture on the structure of local derivations proposed by Ayupov, Elduque, and Kudaybergenov (J. Pure Appl. Algebra, 2023), as well as the conjecture concerning the structure of almost inner derivations proposed by Burde, Dekimpe, and Verbeke (J. Algebra Appl., 2018).

We classify all Rota-Baxter operators of weight $\lambda$ on $M_n(F)$ over a field $F$ of characteristic zero whose minimal polynomial has maximal degree and show that in both cases $\lambda = 0$ and $\lambda \neq 0$ such an operator is conjugate to exactly one operator.

We study super-matrices $M_{m|n}(R)$ over the rank-two exterior algebra $R=K[\theta_1,\theta_2]/(\theta_1^2,\theta_2^2,\theta_1\theta_2+\theta_2\theta_1)$, the smallest supercommutative ring on which the Berezinian's odd-by-odd correction term $BD^{-1}C$ need not vanish. We recall that the supertrace satisfies graded cyclicity, $\mathrm{str}(XY)=(-1)^{|X||Y|}\mathrm{str}(YX)$, so $\mathrm{str}([A,B])=0$ for even $A,B$, and that an explicit $1|1$ matrix over $R$ has $BD^{-1}C\neq0$ surviving to second order in the odd generators. We then prove the extension anticipated in the $1|1$ note: a closed-form Berezinian formula for arbitrary even $X\in M_{m|n}(R)$, $m,n\geq1$ (Theorem 1), given through a single $m\times m$ matrix pairing $\Delta$ built from the odd blocks of $X$; and a fully general multiplicativity theorem $\mathrm{Ber}(XY)=\mathrm{Ber}(X)\mathrm{Ber}(Y)$ for even invertible $X,Y\in M_{m|n}(R)$ (Theorem 2), proved by an explicit trace-cyclicity cancellation. Both results specialize exactly to the known $1|1$ formulas when $m=n=1$, and we verify them on a worked $m=2,n=1$ example. We extend the framework from two to an arbitrary number $r\geq2$ of odd generators, replacing the scalar pairing $\Delta$ by a $\Lambda^2(V)$-valued pairing on the $r$-dimensional space $V$ of odd generators, and identify this pairing, in the scalar case, with a Plücker coordinate of the odd off-diagonal data. We close with a discussion of related work of Khudaverdian–Voronov, and a remark identifying $\Delta$ as a matrix-valued contraction against the natural alternating pairing on the space of odd generators.

Motivated by the Casas-Loday-Pirashvili maps concerning representations of $n$-Leibniz algebras, we give an intrinsic characterization of the cohomology of $n$-Lie algebras with coefficient in a representation space $V$. We investigated the cohomology theory of $n$-Lie algebras in three different ways: cohomology theory of Leibniz algebras, infinitesimal deformation and abelian extension. In the main part of this paper, we solve the extending problems for $n$-Lie algebras. A necessary and sufficient unified-product criterion is obtained. The case of crossed products, sparse non-abelian extensions, matched pairs are studied as special cases. We also investigate the factorizations, deformation maps, and complements problem for $n$-Lie algebras. An appendix removes the general non-reduced unified product with all intermediate mixed components.

We determine the ozone group of \(U_q^{+}(B_2)\). More precisely, if \(\ell=\operatorname{ord}(q^2)\) when \(q\) is a root of unity, we obtain $$ \operatorname{Oz}(U_q^{+}(B_2)) \cong \begin{cases} \mu_2 , & \text{if \(q\) is not a root of unity},\\[2mm] \mu_{\gcd(\ell,2)}, & \text{if \(q\) is a primitive \(m\)-th root of unity, \(m\geq5\).} \end{cases} $$ We also study several homological properties of \(U_q^{+}(B_2)\), showing that it is Artin–Schelter regular of global dimension \(4\), Auslander-regular, Cohen–Macaulay, strongly Noetherian, and skew Calabi–Yau. Using two different iterated Ore presentations, we show that its Nakayama automorphism is trivial for every $q \in \K^*$, so $U_q^{+}(B_2)$ is Calabi–Yau of dimension $4$; for $q$ not a root of unity this also follows from work of Goodearl and Yakimov. When $q$ is a primitive $m$-th root of unity with $m\geq5$, we deduce that the center is Gorenstein, we check directly that the ozone group acts with trivial homological determinant, and we describe the nonzero normal elements in terms of the ozone group.

Let $K$ be a field and let $\T=L_K(E_T)$ be the algebraic Toeplitz algebra. We classify the Lie ideals of $\T^-$. If $w$ is the sink of the Toeplitz graph and $F=I(w)$, then $F\cong M_\infty(K)$, $\T/F\cong K[t,t^{-1}]$, and $[\T,\T]=F$. We prove that every noncentral Lie ideal contains $\mathfrak s=[F,F]$, the finitary trace-zero Lie algebra. Thus the classification reduces to the Heisenberg-type quotient $\T/\mathfrak s$. The Lie ideals are precisely $0$, $K1$, the spaces $\mathfrak s+U$, where $U\leq P_K\oplus Kw$ and $w\notin U$, and the inverse images $\rho^{-1}(W)$, where $W\leq K[t,t^{-1}]$. Here $P_K=K1$ in characteristic $0$, and $P_K=\operatorname{span}_K\{1,e^{m\ell},(e^*)^{m\ell}:m\geq1\}$ if $\operatorname{char}K=\ell>0$. We also describe the Lie derivations of $\T^-$ as sums of associative derivations of $\T$ and central Lie derivations factoring through $\T/F$.

The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over \(\mathbb F_{2^n}\), with field elements expressed in varying ordered \(\mathbb F_2\)-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size \(n\), the correspondence between ordered bases and distinct inversion maps is exactly \(n\)-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree \(n-1\) and joint ANF leap \(1\), the mixed representation formulation has degree \(2(n-1)\) and joint ANF leap \(2\), and the complete raw formulation has degree at most \(3(n-1)\) and joint ANF leap at least \(n\). Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.

A finite simple graph $G$ is called a cograph if it does not contain the path on four vertices $P_4$ as an induced subgraph. It is classically known that the family of cographs are well-quasi-ordered by the induced subgraph relation \cite{D}. In preceding work of Knudsen and the third author \cite[Theorem 7.2]{KR}, it was shown that this well-quasi-order statement admitted a categorification, which allowed those authors to prove universal finite generation statements about the homology groups of configuration spaces on cographs \cite[Theorem 1.5]{KR}. In this work, we expand \cite[Theorem 7.2]{KR} to be compatible with the family of polynomial rings on the vertex sets of cographs. By consequence, we are able to prove a number of universality results related with edge and toric ideals of these polynomial ring, partially generalizing and expanding upon work of Kahle \cite{kahle2019binomial}. We also conclude strong restrictions on the kinds of topologies that can arise from graph complexes and anchored configuration spaces associated to cographs, as well as combinatorial constraints on the possible combinatorics of hyperplane arrangements of cographs.

The factorization problem is a classical topic in the theory of polynomial rings. In this paper, we obtain an irreducibility criterion for central polynomials over division algebras. An example demonstrating the computation of factors for a central polynomial is also presented.

We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. We prove that the good brackets can be steered simultaneously and arbitrarily in small time, which yields sufficient conditions for the small-time local controllability of control-affine systems. This positive result draws on constructions of Kawski, Agrachev-Gamkrelidze and Krastanov. Conversely, we prove that each bad bracket generates an obstruction to controllability, hence a family of necessary conditions.

Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.

The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik–Solomon algebra. It is known that the defining ideal of the Orlik–Solomon algebra has a quadratic Gröbner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik–Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik–Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over $\mathbb{Q}$, and of any rank $\geq 3$. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik–Terao algebras coming from non-supersolvable arrangements.

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

For categories, being balanced (meaning that every arrow that is both epic and monic is an isomorphism) can be regarded as a strong tameness property which plays an important role, for example, in algebra and logic. We study the problem of associating a balanced companion category, called a Beth companion, to a locally finitely presentable category (equivalently, the category of models of a finitary essentially algebraic theory). We show that, if it exists, the Beth companion is unique and can be described in terms of saturated objects. Under additional assumptions, we prove that Beth companions can be computed as orthogonality classes, and admit a syntactic presentation via Gabriel–Ulmer duality. Furthermore, we establish conditions for the transfer of properties from a category to its Beth companion. A number of examples of categories that do or do not admit a Beth companion are provided. In particular, we show that the category of commutative rings does not admit a Beth companion, and employ Pierce sheaf representations to describe the Beth companion of other classes of rings.

We modify the definition of eventually homological isomorphisms so that it becomes more natural and fruitful. We introduce singular equivalences of $n$-adjoint type which can be viewed as standard singular equivalences of different levels and are more convenient for reducing various homological conjectures and transferring homological properties. We develop the theories of singular equivalences of $n$-adjoint type for $n\ge 2$ and standard eventually homological isomorphisms between derived categories by giving their characterizations in terms of $\otimes$-(co)perfect complexes of bimodules, two constructions via opposite algebras and tensor product algebras, examples induced by homomorphisms of algebras and idempotents, and applications to the reduction of homological conjectures such as the finitistic dimension conjecture, Gorenstein symmetry conjecture, Auslander-Reiten conjecture and Gorenstein projective conjecture, as well as to the preservation of homological properties such as Gorensteinness, the Fg condition, Happel's property and Han's property. Moreover, we show that standard eventually homological isomorphisms correspond to fully faithful standard triangle functors between singularity categories with canonical left adjoint, and essentially surjective standard eventually homological isomorphisms between derived categories correspond to singular equivalences of 3-adjoint type, thereby incorporating essentially surjective standard eventually homological isomorphisms into the framework of singular equivalences of 3-adjoint type.

In this paper, we mainly investigate $n$-cotorsion pairs in two types of abelian categories. Firstly, we construct left (right) $n$-cotorsion pairs in comma categories by improving some isomorphisms between homology groups and combining some classes of objects in comma categories. Secondly, based on known homological formulas over trivial ring extensions, we study some classes of modules satisfying certain conditions over such rings, and we establish left (right) $n$-cotorsion pairs over trivial ring extensions through these classes. Then corresponding conclusions over Morita rings with zero bimodule homomorphisms are directly presented. The hereditary property of left (right) $n$-cotorsion pairs is investigated in both cases. Finally, we apply main results obtained in comma categories and over trivial ring extensions to formal triangular matrix rings. At the same time, via these applications, we found that relevant conditions in known results have been improved and the known results have been elevated.

Let $A_\bullet$ and $B_\bullet$ be nonnegatively graded algebras, finitely generated in degrees less than $2$ and with semisimple base rings $A_0$ and $B_0$. We prove that if $A \simeq B$ as ungraded algebras, then $A_\bullet \simeq B_\bullet$ as graded algebras. This generalises a theorem of Bell and Zhang arXiv:1509.08812 in the connected case $A_0 = k = B_0$, and the path-algebra version when $A_0$ and $B_0$ are $k$-elementary due to Gaddis arXiv:1712.01650. We also discuss some related open problems for graded algebras.

We consider in this paper complex Lie groups $\mathrm{SL}_n(\mathbb{C})$ for $n\geq 2$, $\mathrm{SO}_n(\mathbb{C})$ for $n\geq 5$, and $\mathrm{Sp}_{2n}(\mathbb{C})$ for $n\geq 1$. We show that for any such Lie group, the Lie algebra of polynomial vector fields can be generated by three $\mathbb{C}$-complete polynomial vector fields.

The paper considers the generalization of the construction of orthogonal completion to all semiprime linear algebras (systems), its implementation in maximal systems of quotients similar to the symmetric Martindale ring of quotients for known examples of their description in terms of algebras of quotients of graded Lie algebras and technical results for applying the orthogonal completion method to them.

Motivated by the recent study of strongly $\Delta$-clean rings, we introduce and study strongly $m$-$\Delta$-clean rings. We establish their fundamental properties and characterize strongly $m$-$\Delta$-clean rings via lifting $m$-potent elements modulo $\Delta(R)$. We prove that every $m$-$\Delta$-clean ring is clean and that the factor ring of a strongly $m$-$\Delta$-clean ring modulo its Jacobson radical is reduced. Finally, we investigate corner rings, Morita context rings, and the relationships of these rings with $\Delta$-clean, local, semipotent, and strongly $m$-nil clean rings.

Let $R$ be a commutative $\Q$-algebra and let $HR$ denote its Hurwitz series ring. We introduce a one-parameter convolution $\hconv{t}$ on $HR$ for which the specialization $t=-1$ is the intrinsic Hurwitz product. Positive integral parameters act on finite-support truncations and reproduce finite free convolution in coefficient coordinates. A logarithmic transform yields additive and homogeneous cumulants, an explicit inversion formula, binomial, Hermite, Laguerre, and hypergeometric families, and coefficientwise analogues of the law of large numbers and the central limit theorem. At $t=-1$, addition of independent random variables becomes multiplication of their moment sequences in $HR$; this gives applications to classical cumulants, beta–gamma products, and self-decomposable laws. Weighted coefficient norms turn $\hconv{t}$ into a commutative Banach algebra product. Norm-continuous convolution semigroups then have generators conjugate to multiplication operators. Explicit formulas are obtained for the Hermite and Laguerre semigroups, the finite free heat generator, and the Lévy–Khintchine generator.

We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This generalizes and unifies a number of results to the effect that quantum groups have only classical symmetries, due to Kasprzak et al., Budziński-Kasprzak and Brannan et al. The stated automatic classicality follows from the fact that the antipode, counit and tensorand-interchange of a Hopf algebra internal to any symmetric monoidal category are contained in the (non-symmetric) monoidal subcategory generated by the associative-algebra structure and the comultiplication, assuming a weak form of arrow-retraction invariance.

Global representations are subdirect representations satisfying a sheaf-like local-to-global patching principle. We develop a unified and simplified theory of such representations. For quasivarieties with a near-unanimity term, this framework recovers the main classical representation theorems through a common argument and yields a general representation by factors of bounded subdirect width. When the relatively subdirectly irreducible members form a universal class, we obtain an optimal result: every relatively congruence-distributive algebra admits a global representation by relatively globally indecomposable factors, which we characterize explicitly. We also provide a converse: global representations by factors of bounded relative subdirect width force the existence of a near-unanimity term. Adding semisimplicity to the hypotheses of the main theorem yields optimal representation results for filtral quasivarieties and dual discriminator varieties, together with a simple description of the globally indecomposable algebras. The technical engine behind these results is a new infinitary extension of the congruence-system component of the Baker–Pixley theorem.

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from $\mathfrak{S}_{N}$ to $\mathfrak{S}_{N-1}$. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the Bratelli diagram of a family of algebras $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ by Okada in 1994. In this paper, we first realize the Okada algebra $\mathbf{O}_N(X,Y)$ and the associated monoid $\mathbf{O}_N$ using a labelled version of non-crossing arc-diagrams appearing in the description of the Temperley-Lieb algebra and Jones monoid. We establish, for general parameters $(X,Y)$, that the dimension of the Okada algebra $\mathbf{O}_N(X,Y)$ is $N!$, noting that Okada proved this result only in the semisimple case. We interpret a natural bijection between permutations and labelled arc-diagrams as an incarnation of Fomin's version of the Robinson-Schensted correspondence associated to the Young-Fibonacci lattice. The arc-diagram formalism allow us to probe the structure of the Okada monoid and algebra. In particular we prove that the Okada monoid is a regular, aperiodic $*$-monoid and we describe its Green relations and order. These results allow us to construct a cellular basis of the Okada algebra and to show that $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ forms a coherent tower of cellular algebras in the sense of Goodman and Graber. We present some conjectures expressing the Gram determinant of the invariant bilinear form attached to each cell module in terms of Okada's clone Schur functions. We conclude the paper by presenting two follow-up, ongoing projects along with a series of questions pushing further the analogy between the symmetric groups and the Okada algebras.

We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford–Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer–Cartan form is shown to decompose into a direct sum of the Maurer–Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer–Cartan form. Building on this correspondence, we introduce and develop the notion of quantum $\textrm{G}$-structure, proving that quantum $\textrm{G}$-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter–Drinfeld module of $O_q(\mathrm{GL}_2)$.

In this paper, we introduce the controlling $L_{\infty}[1]$-algebra of extended Rota-Baxter algebras. As applications, we obtain the cohomology of extended Rota-Baxter algebras and homotopy extended Rota-Baxter algebras. We also study abelian extensions and formal deformations by the lower cohomology group. Lastly, we describe the intrinsic relationship among extended Rota-Baxter algebras, extended Rota-Baxter Lie algebras, and the eigenvalues of the extended Rota-Baxter operator. As the generalization of Rota-Baxter algebras with weight and modified Rota-Baxter algebras, the corresponding results also hold for those algebras.

Let $R$ be a finite associative ring with identity, and let $\Nul(R)$ denote the set of polynomials in a central indeterminate that vanish identically on $R$ under right evaluation. We establish two results concerning the failure of $\Nul(R)$ to be a right ideal. First, if $e$ is an idempotent, $f=1-e$, and either $eRf=0$ or $fRe=0$, then right multiplication by $e$ preserves $\Nul(R)$. Thus both off-diagonal Peirce components must be nonzero whenever $e$ witnesses a failure of right stability. Second, if $J$ is a finite nilpotent ideal whose additive group is a $2$-group and $e,f$ are complementary idempotents such that \[ eJf\neq0,\qquad fJe\neq0,\qquad J^3\neq0, \] then $|J|\geq32$. Werner's theorem that $\Jac(R)^3=0$ implies the two-sidedness of $\Nul(R)$ therefore yields $|R|\geq128$ whenever $\Nul(R)$ is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.

Let $K$ be any field, $E$ any directed graph, and $L_K(E)$ the associated Leavitt path algebra. As described first by Chen, and subsequently generalized by Ara and Rangaswamy, for each cycle $c$ in $E$ one can build the simple left $L_K(E)$-module $V_c^E$, and then more generally $V_{p(x),c}^E$ (where $p(x)$ is an irreducible polynomial in $K[x,x^{-1}]$). A cycle $c$ is called exclusive in case none of the vertices of $c$ is the base of any cycle other than $c$. In our main result we provide an explicit description of the injective envelope of $V_c^E$, and then more generally of $V_{p(x),c}^E$, for each exclusive cycle $c$. Our method involves defining an $L_K(E)$-module structure on an appropriately-built $K$-vector space of infinite series. Our main result significantly generalizes previous work of the authors, in that the result holds for all graphs (finite or not), and all exclusive cycles.

Let $K$ be any field and $E$ any directed graph. We characterize up to isomorphism the simple (i.e., minimal) left ideals of the Leavitt path algebra $L_K(E)$. Then, for each simple %(i.e., minimal) left ideal $I$ of %the Leavitt path algebra $L_K(E)$, we explicitly construct the injective envelope of $I$. This result generalizes to all graphs $E$ and all simple left ideals in $L_K(E)$ the construction presented previously by the three authors for the specific case of the Jacobson algebra $R=K\langle X,Y | XY=1\rangle$ and the simple left $R$-ideal $R(1-YX)$. Our method involves defining an $L_K(E)$-module structure on a $K$-vector space of infinite series. We conclude the article by showing how our construction directly gives a description of the injective envelope of simple $L_K(E)$-modules arising from two types of infinite emitters in $E$.

Let $R\ltimes_{n}M$ be the $n$-trivial extension of a ring $R$ by an $R$-bimodule $M$. We first characterize projective, injective, flat and finitely generated modules over $R\ltimes_{n}M$. Then we describe when the ring $R\ltimes_{n}M$ is left perfect (Noetherian, coherent, Artinian, hereditary, Kasch). Finally, we establish some homological formulas of $n$-trivial extensions of rings.