2021/04/15 by Vesselin Drensky · 1 citation
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Algebraic structures and combinatorial models
paper · doi:10.1080/03081087.2021.1910121
Let A1,…,As be unitary commutative rings which do not have non-trivial idempotents and let A=A1⊕⋯⊕As 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 an arbitrary set of variables X. We apply this result to the matrix ring M2(Zn[[X]]) over the ring Zn[[X]] where Zn≅Z/nZ 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=Zn, the Chinese remainder theorem and the Euler-Fermat theorem.