Euclidean domain

In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain. It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural

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In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonz

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an

An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the correspond

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On finitely generated modules over quasi-Euclidean rings

Luc Guyot

Let R be a unital commutative ring, and let M be an R-module that is generated by k elements but not less. Let be the subgroup of generated by the elementary matrices. In this paper we study the action of by matrix multiplication on the set of unimodular rows of M of length . Assuming R is moreover Noetherian and quasi-Euclidean, e.g., R is a direct product of finitely many Euclidean rings, we show that this action is transitive if . We also prove that is equipotent with the unit group of where is the first invariant factor of M. These results encompass the well-known classification of Nielsen non-equivalent generating tuples in finitely generated Abelian groups.

Springer Verlag2017

On quotients of generalized Euclidean group rings

Luc Guyot

2018

We use the averaged variational principle introduced in a recent article on graph spectra [10] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kroger's bound for Neumann spectra of Laplacians on Euclidean domains [15]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces ( here extending a result of Strichartz [26] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schrodinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

European Mathematical Soc2017

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In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonz

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an