Idempotents of 2× 2 matrix rings over rings of formal power series

2020/06/26 by Vesselin Drensky, Drensky, Vesselin
Mathematics · #13B25 #13F25 #16S50 #16U40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2006.15070

openalex publication_date 2020/06/26 · openalex created_date 2020/07/02 · openalex updated_date 2026/07/28

Abstract

Let A₁,…,A_s be unitary commutative rings which do not have non-trivial idempotents and let A=A₁⊕⋯⊕ A_s be their direct sum. We describe all idempotents in the 2× 2 matrix ring M₂(A[[X]]) over the ring A[[X]] of formal power series with coefficients in A and in arbitrary set of variables X. We apply this result to the matrix ring M₂(\mathbb Z_n[[X]]) over the ring \mathbb Z_n[[X]] for an arbitrary positive integer n greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for 2× 2 matrices only) and, in the special case A=\mathbb Z_n, the Chinese reminder theorem and the Euler-Fermat theorem.

Citations

• Arithmetic of idempotents in ℤ/m ℤ
• Effectively bounded idempotent generation of certain 2 \× 2\n singular matrices by idempotent matrices over real quadratic number rings
• Primitive idempotents in central simple algebras over \mathbbF_q(t) with an application to coding theory

Related

• Idempotents of 2 × 2 matrix rings over rings of formal power series
• Matrices over a commutative ring as sums of three idempotents or three involutions
• On power sums of matrices over a finite commutative ring
• Commutators of finite multiplicative order
• Triangular idempotent matrices over a general ring