On November 23, 2020 (Monday) at 11:15 a.m. (local time in Bulgaria) a joint Zoom-webinar will be held of the Algebra and Logic Seminar of IMI – BAS and the Algebra Seminar of the Alfréd Rényi Institute of Mathematics, Budapest, Hungary. Two talks on:

I. Anniversary: 150 Years of idempotents.
II. Idempotents of 2 × 2 matrix rings over rings of formal power series.

will be delivered by Vesselin Drensky.

Join the Zoom meeting at

https://zoom.us/j/95342234576?pwd=TTFrZmwyQ2h1QmhOc1JaWXorbzI4dz09

Meeting ID: 953 4223 4576
Passcode: 943760

Abstract. An element a in a ring A is called an idempotent if a2=a. In 2020 we celebrate an anniversary of the idempotents –150 years of their discovery. The idempotents were introduced in Ring Theory by Benjamin Peirce in 1870. Already 150 years their study is among the main topics in Ring Theory and its applications. The first part of the talk is devoted to the story of the discovery of the idempotents.
The second part of the talk surveys some results on idempotents of matrix rings over commutative unitary rings. We present also a new result from V. Drensky, Idempotents of 2 × 2 matrix rings over rings of formal power series, arXiv:2006.15070v1 [math.RA]

This is the description of idempotents of M2(A[[X]]), where A is a direct sum of a finite number of commutative rings without non-trivial idempotents and A[[X]] is the ring of formal power series in an arbitrary (also infinite) set of commuting variables. As a consequence we describe the idempotents of M2(Zn[[X]]) when n is an arbitrary positive integer greater than 1.

Our proofs are very transparent and use well known elementary arguments only. They are based on the Cayley-Hamilton theorem (for 2 × 2 matrices only), the Chinese remainder theorem and the Euler-Fermat theorem.
От секция „Алгебра и логика” на ИМИ – БАН
http://www.math.bas.bg/algebra/seminarAiL/