Let k be an integral domain, n a positive integer, X a generic n×n matrix over k (i.e., the matrix (xij) over a polynomial ring k[xij ] in n 2 indeterminates xij), and adj(X) its classical adjoint. For char k = 0 it is shown that if n is odd, adj(X) is not the product of two noninvertible n×n matrices over k[xij ], while for n even, only one particular sort of factorization can occur. Whether t...

In this note the authors correct and extend results presented in their article “The Isomorphism problem for incidence rings”, Pacific J. Math., 187(2) (1999), 201-214. Specifically, it is shown that for a large class of rings (including those with finite right Goldie dimension, semilocal, and many commutative rings), if P and P ′ are finite preordered sets for which there is an isomorphism of i...

A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodro...

Let k be a field. Then Gaussian elimination over k and the Euclidean division algorithm for the univariate polynomial ring k[x] allow us to write any matrix in SLn(k) or SLn(k[x]), n ≥ 2, as a product of elementary matrices. Suslin’s stability theorem states that the same is true for the multivariate polynomial ring SLn(k[x1, . . . , xm]) with n ≥ 3. As Gaussian elimination gives us an algorith...

1. Amitsur in his paper on Finite Dimensional Central Division Algebras [l] has proved that in a division ring D with center C, (P: C) 5= ra2 < =o if and only if every primitive homomorphic image of a polynomial ring P[x] is a complete matrix ring Ah, h^n, over a division ring A. Equivalently speaking, a division ring is finite dimensional over its center if and only if the polynomial ring over...

The class of Bézout factorial rings is introduced and characterized. Using the factorial properties of such a ring R, and given a n×m matrix A over R, we find P ∈ GL(n, R) and Q ∈ GL(m, R) such that PAQ is diagonal with every element in the diagonal dividing the following one. Key-words: Ring, Bézout, principal, factorization, reduction of matrices.

Robert Lee Wilson Department of Mathematics Rutgers University The determinant of a matrix with entries in a commutative ring is a main organizing tool in commutative algebra. In these lectures, we present an analogous theory, the theory of quasideterminants, for matrices with entries in a not necessarily commutative ring. The theory of quasideterminants was originated by I. Gelfand and V. Retakh.

we introduce center-like subsets z*(r,f), z**(r,f) and z1(r,f), where r is a ring and f is a map from r to r. for f a derivation or a non-identity epimorphism and r a suitably-chosen prime or semiprime ring, we prove that these sets coincide with the center of r.

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

let r be a prime ring with extended centroid c, h a generalized derivation of r and n ⩾ 1 a xed integer. in this paper we study the situations: (1) if (h(xy))n = (h(x))n(h(y))n for all x; y 2 r; (2) obtain some related result in case r is a noncommutative banach algebra and h is continuous or spectrally bounded.