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Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic solutions to the Hom-Yang-Baxter equation (HYBE for short). This paper investigates set-theoretic solutions to HYBE and associated algebraic system, called Hom-cycle sets. We characterize left non-degenerate involutive set-theoretic solutions to HYBE and Hom-cycle sets, and establish their relations. We discuss connections among Hom-cycle sets, cycle sets, left non-degenerate involutive set-theoretic solutions to HYBE and the Yang-Baxter equation.

Let K be a commutative algebra over a field and A=K[x1,x2,…,xn] the polynomial algebra over K. It is proved that the discriminant of A over the symmetric polynomials is the Vandermonde determinant ...

We present conditions for a set of matrices satisfying a permutation identity to be simultaneously triangularizable. As applications of our results, we generalize Radjavi’s result on triangularization of matrices with permutable trace and results by Yan and Tang on linear triangularization of polynomial maps.

Let F be a polynomial map in two variables over the complex field with nonzero constant Jacobian. Abhyankar proved that if F is smooth on generic lines then F is invertible. We sharpen Abhyankar’s result by proving that if F(𝕃) is smooth on one line 𝕃 then F is invertible. Criterion for smoothness of F(𝕃) is also discussed.

To generalize D-nilpotent matrices that play a role in study of Druzkowski maps, we introduce quasi-D-nilpotent matrices. A matrix A is called quasi-D-nilpotent if there exists a subspace V of diagonal matrices of codimension 1 such that DA is nilpotent for all . It is proved that a quasi-D-nilpotent matrix has few nonzero principal minors. We also determine irreducible quasi-D-nilpotent matrices and the Frobenius normal forms of quasi-D-nilpotent matrices with respect to permutation similarity.

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.

In this paper expressions for the Drazin inverse of a modified matrix A - CDdB are presented in terms of the Drazin inverses of A and the generalized Schur complement D - BAdC under weaker restrictions. Our results generalize and unify several results in the literature and the Sherman-Morrison- Woodbury formula.

Let k ⊆ k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x1,…, xn] denotes the polynomial ring in n variables over the field k. More precisely, let D = {Dn}n=0∞ a higher k-derivation on k[X] and D′ = {D′n}n=0∞ a higher k′-derivation on k′[X] such that D′m(xi) = Dm(xi) for all m ⩾ 0 and i = 1, 2,…, n. Then (1) k[X]D = k if and only if k′[X]D′ = k′; (2) k[X]D is a finitely generated k-algebra if and only if k′[X]D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X]D of a higher derivation D of k[X] can be generated by a set of closed polynomials.

Denote by \({\mathcal{C}\ell_{p,q}}\) the Clifford algebra on the real vector space \({\mathbb{R}^{p,q}}\). This paper gives a unified tensor product expression of \({\mathcal{C}\ell_{p,q}}\) by using the center of \({\mathcal{C}\ell_{p,q}}\). The main result states that for nonnegative integers p, q, \({\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}\) where \({p + q \equiv \varepsilon}\) mod 2, \({\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}\) mod 8 and \({\delta = \lfloor i / 4 \rfloor}\).

An endomorphism $\varphi$ of a polynomial ring is called preserving outer rank if $\varphi$ sends each polynomial to one with the same outer rank. For the polynomial ring in two variables over a field of characteristic $0$ we prove that an endomorphism $\varphi$ preserving outer rank is an automorphism if one of the following conditions holds: (1) The Jacobian of $\varphi$ is a non-zero constant; (2) The image of $\varphi$ contains a coordinate; (3) $\varphi$ has a ``fixed point". DOI: 10.1017/S0004972712000044

Let 𝕂 be a field and r 1, …, r t positive integers. Let r 1# ··· #r t be the smallest positive integer n for which there exists a non-singular multilinear map f : 𝕂 r 1 × ··· × 𝕂 r t → 𝕂 n . We show that for any field 𝕂, and the equality holds if 𝕂 is an algebraically closed field. As a consequence, we settle a problem on linear dependence of multilinear forms proposed by Aron et al. [R. Aron, L. Downey, and M. Maestre, Zero sets and linear dependence of multilinear forms, Note Mat. 25 (2005), pp. 49–54]. When 𝕂 = ℝ, we prove that r#r ⩽ 2r − 2 for any r > 1, which is also applied to sharpening the lower bound of the maximal dimension of a subspace of matrices without rank-one matrices.

Let f=x+H:Cn→Cn be a homogeneous polynomial map of degree d⩾2 and FA=y+(Ay)(d):CN→CN a power linear map such that f and FA are a generalized Gorni-Zampieri pair. We discuss the relation between the nilpotency indices of JH and J(Ay)(d) and we show that f is linearly triangularizable if and only if FA is linearly triangularizable. As a consequence, we show that a quadratic linear Keller map FA=y+(Ay)(2) with nilpotency index three, i.e., (J(Ay)(2))3=0, is linearly triangularizable.

For any n × n complex matrix A and any integer k ⩾ 1, we prove that the span of image of the map x ↦ ( Ax ) ( k ) is equal to the range of ( AA ∗ ) ( k ) , where X ∗ and X ( k ) denote the conjugate transpose and the k th Hadamard power of a matrix X , respectively. This settles a conjecture, due to Gorni and Tutaj-Gasinska, related to the study of the Jacobian Conjecture.

For a ring R, denote by the circle multiplication of R defined by for any . Then is a semigroup with the identity 0, called the adjoint or circle semigroup of R. There is an interest in the relatio...

A ring R is called left T-idempotent (left T-stable) if for any sequence r 1,r 2,… in R there exists a positive integer n such that r 1 r 2… rn is an idempotent (r 1 r 2… rn = r 1 r 2… r n+1 = …). In this paper we characterize left T-idempotent rings and left T-stable rings and describe left T-idempotent semigroup rings and left T-stable semigroup rings.

Let $3\leq d_1\leq d_2\leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $\frac{d_1}{\gcd(d_1,d_3)}\neq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$; (2) If $d_3$ is a prime number and $\gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.

Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $a\mid 2d$, then $(a, a+d, a+2d)\in\mdeg(\Tame(\mathbb{C}^3))$. (2) If $a\nmid 2d$, then, except for arithmetic progressions of the form $(4i,4i+ij,4i+2ij)$ with $i,j \in\mathbb{N}$ and $j$ is an odd number, $(a, a+d, a+2d)\notin\mdeg(\Tame(\mathbb{C}^3))$. We also related the exceptional unknown case to a conjecture of Jie-tai Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials.

Let F : C n → C m be a polynomial map with deg F = d ≥ 2 . We prove that F is invertible if m = n and ∑ i = 1 d − 1 ( J F ) | α i is invertible for all α i ∈ C n , which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines L = { β + μ γ ∣ μ ∈ C } ⊆ C n ( γ ≠ 0 ), F ∣ L is linearly rectifiable, if and only if ∑ i = 1 d − 1 ( J F ) | α i ⋅ γ ≠ 0 for all α i ∈ L . This appears to be the case for all affine lines L when F is injective and d ≤ 3 . We also prove that if m = n and ∑ i = 1 n ( J F ) | α i is invertible for all α i ∈ C n , then F is a composition of an invertible linear map and an invertible polynomial map X + H with linear part X , such that the subspace generated by { ( J H ) | α ∣ α ∈ C n } consists of nilpotent matrices.

To investigate the generalized Drazin invertible of a 2 × 2 operator matrix M, representations for the generalized Drazin inverse of M in terms of its individual blocks are presented under some conditions and some recent results are extended.

It is shown that the nth center of a radical ring coincides with that of its adjoint group, from which a result of Jennings is sharpened and a conjecture of his is confirmed.

It is proved that a ring is periodic if and only if, for any elementsxandy, there exist positive integersk,l,m, andnwith eitherk≠morl≠n, depending onxandy, for whichxkyl=xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and aJ-ring.

We prove that n pairwise commuting derivations of the polynomial ring (or the power series ring) in n variables over a field k of characteristic 0 form a commutative basis of derivations if and only if they are k -linearly independent and have no common Darboux polynomials. This result generalizes a recent result due to Petravchuk and is an analogue of a well-known fact that a set of pairwise commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector.

Let Bl, l = 1, … , k, be m × nl complex matrices and let x [ l ] ∈ C n l , l = 1 , … , k , be complex vector variables. We show that the components of the map H = ( B 1 x [ 1 ] ) ( d 1 ) ∘ ⋯ ∘ ( B k x [ k ] ) ( d k ) are linearly dependent over C if and only if det ( B 1 B 1 ∗ ) ( d 1 ) ∘ ⋯ ∘ ( B k B k ∗ ) ( d k ) = 0 , where ∘ means the Hadamard product, X ∗ and X ( d ) denote the conjugate transpose and the dth Hadamard power of a matrix X respectively. Connections are established between the Homogenous Dependence Problem (HDP( n , d )), which arises in the study of the Jacobian Conjecture, and the dependence problem for power linear maps (PLDP( n , d )). An algorithm is given to compute counterexamples to PLDP( n , d ) from those to HDP( n , d ), and counterexamples to PLDP( n , 3 ) are obtained for all n ⩾ 67 .