Open AccessThe existence of clean elements in a matrix ring over integral domain and its connections with g(x)-cleanness and strongly g(x)-cleanness
Author(s) -
IF Ambarsari,
Santi Irawati,
I Made Sulandra,
Hery Susanto,
Angelina Chin Yan Mui,
Hidetoshi Marubayashi
Publication year - 2019
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1397/1/012069
Subject(s) - integral domain , mathematics , idempotence , element (criminal law) , unit (ring theory) , subring , ring (chemistry) , domain (mathematical analysis) , matrix (chemical analysis) , polynomial ring , combinatorics , pure mathematics , polynomial , mathematical analysis , chemistry , materials science , composite material , mathematics education , organic chemistry , field (mathematics) , political science , law
An element a in a ring R with unity is called clean, if there exist an idempotent element e ∈ R and a unit element u ∈ R such that a = e + u . This article aims to show all of clean elements in a certain subring X 3 ( R ) of a matrix ring 3 × 3 over integral domain R and their connections with g ( x )-cleanness and strongly g ( x )-cleanness for some fixed polynomial g ( x ). To achieve it, we found out unit and idempotent elements in X 3 ( R ) for constructing clean elements and selected some fixed g ( x ) in the center of R for investigating their relations with g ( x )-cleanness and strongly g ( x )-cleanness. In this article, we obtained eight general forms of the clean elements in X 3 ( R ), g ( x )-clean elements with g ( x ) = x n − x , which five forms of them were strongly g ( x )-clean but the other three forms were not. The latter result was shown by providing counter examples.