[Todos CMAT] SEMINARIO DE ALGEBRA Y TEMAS AFINES EL PROXIMO LUNES

[Walter Ferrer]

Seminario de álgebra y temas afines
Centro de Matemática: Sala de seminarios del piso 14
Hora: 1430/16 --luego haremos un modesto brindis--
Día: Lunes 13 de julio de 2015
Título: "Euclidean pairs, decomposition into idempotents and related topics"

Resumen :

An ordered pair (a,b) in any ring R is said to be a
right Euclidean pair if there exist elements (q_{1},r_{1}), ....,

(q_{n+1},r_{n+1}) en R^{2} (for some n\geq 0)

such that a=bq_{1}+r_{1}, b=r_{1}q_{2}+r_{2}, and

r_{i-1}=r_iq_{i+1}+r_{i+1} for 1<i\leq n, r_{n+1}=0.



If  all pairs (a,b)\in R^2 are

right Euclidean, we say that R is a right quasi-Euclidean

ring.  A nice class of quasi euclidean rings is the class of unit

regular rings.


We study the interplay between the classes of right quasi-Euclidean

rings and right K-Hermite rings, and relate them to projective-free

rings and Cohn's GE_2-rings using the method of noncommutative

Euclidean divisions and matrix factorizations into idempotents.

Right quasi-Euclidean rings are closed under matrix extensions, and

a left quasi-Euclidean ring is right quasi-Euclidean if and only if

it is right B\'ezout.   Singular matrices over left and right

quasi-Euclidean domains are shown to be products of idempotent

matrices, generalizing an earlier result of Laffey for singular

matrices over commutative Euclidean domains.


Noncommutative continuant polynomials will naturally appear in

connection with quasi euclidean pairs, and, if time permits, we

will also give some of their properties.

We will try to mention also some more recent works related to

decomposition of nonnegtaive matrices.



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