DOI: https://doi.org/10.14710/jfma.v6i2.19307

IDEMPOTENT ELEMENTS IN MATRIX RING OF ORDER 2 OVER POLYNOMIAL RING $\mathbb{Z}_{p^2q}[x]$

*Muchammad Choerul Arifin orcid  -  Dept. of Mathematics, Gadjah Mada University, Indonesia
Iwan Ernanto  -  Dept. of Mathematics, Gadjah Mada University, Indonesia

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Abstract
An idempotent element in the algebraic structure of a ring is an element that, when multiplied by itself, yields an outcome that remains unchanged and identical to the original element. Any ring with a unity element generally has two idempotent elements, 0 and 1, these particular idempotent elements are commonly referred to as the trivial idempotent elements However, in the case of rings $\mathbb{Z}_n$ and $\mathbb{Z}_n[x]$ it is possible to have non-trivial idempotent elements. In this paper, we will investigate the idempotent elements in the polynomial ring $\mathbb{Z}_{p^2q}[x]$ with $p,q$ different primes. Furthermore, the form and characteristics of non-trivial idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$ will be investigated. The results showed that there are 4 idempotent elements in $\mathbb{Z}_{p^2q}[x]$ and 7 idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$.
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Section: FUNDAMENTAL MATHEMATICS AND APPLICATIONS
Language : EN
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