The next meeting of the Algebra and Logic Seminar will be held on December 2, 2022 (Friday) at 1:00 pm (UTC+2) online in Zoom.
A talk on:
Derivations of upper triangular matrix rings
vs
Derivations of upper triangular matrix semirings
will be delivered by Dimitrinka Vladeva (IMI – BAS).
Abstract. The motivation for this talk is the problem how to represent a derivation of a matrix ring and of an additively idempotent matrix semiring as a sum of well-known derivations.
The results of two of my articles, published in 2022, will be compared and we will draw conclusions about the advantages and disadvantages of these results.
We begin by considering the nature of derivations of triangular matrices over an additively idempotent semiring R generated by left and right semicentral idempotents. Then we construct a semiring D of these derivations and find a basis of D, considered as an R-semimodule. The main result of the first article states that an arbitrary derivation of UTMn(R) (the semiring of upper triangular matrices over an additively idempotent semiring R) is a linear combination of a derivations from the basis of R-semimodule D. When R is an associative ring with identity and UTMn(R) is the ring of upper triangular n x n matrices over R we propose a basis of an additive group D of derivations of UTM_n(R) consisting of derivations δ_i such that δ_i(A) = [e_ii,A], where A ∈UTM_n(R) and e_ii are diagonal matrix units for i = 2, …, n. The main result states that if D is an arbitrary derivation of the ring UTM_n(R) and A ∈UTM_n(R), then there are matrices, such that the derivative D(A) is a linear combination of the values of derivations δ_i ∈D, i = 2, …, n, of these matrices with coefficients the entries of the matrix A.
The seminar will be held online. Join the Zoom link:
https://us02web.zoom.us/j/
Topic: Онлайн семинар на секция “Алгебра и логика”
Time: Dec 02, 2022 01:00 PM Sofia
Meeting ID: 851 3737 5021
Passcode: 035647
Algebra and Logic Department, IMI – BAS
http://www.math.bas.bg/algebra/seminarAiL/
============================== =====================