IDEMPOTENT MATRIX OVER SKEW GENERALIZED POWER SERIES RINGS

Abstract

Let $R[[S,\leq,\omega]]$ be a skew generalized power series ring, with $R$ is a ring with an identity element, $(S,\leq)$ a strictly ordered monoid, and $\omega:S\rightarrow End(R)$ a monoid homomorphism. We define the set of all matrices over $R[[S,\leq,\omega]]$, denoted by $M_{n}(R[[S,\leq,\omega]])$. With the addition and multiplication matrix operations, $M_{n}(R[[S,\leq,\omega]])$ becomes a ring. In this paper, we determine the sufficient conditions for $R$, $(S,\leq)$, and $\omega$, so the element of $M_{n}(R[[S,\leq,\omega]])$ is an idempotent matrix.