Construction of strongly r-clean elements in the certain 3x3 matrix ring over ℤp2, p prime

• Published in: AIP Conference Proceedings 2024-02, Vol.3049 (1)
• Main Authors: Putri, Erlinda Kartika, Irawati, Santi, Marubayashi, Hidetoshi
• Format: Article
• Language: English
• ISSN: 0094-243X; 1551-7616
• DOI: 10.1063/5.0194392

A ring R is called strongly r-clean ring if every element x ∈ R can be written as x = r + e with re = er, for some regular element r and idempotent element e in R. This article aims to show all strongly r-clean elements in the certain ring of matrices T3(ℤp2)={[a11¯0¯0¯0¯a22¯0¯a31¯0¯a33¯]aij¯∈ℤp2;i,j=1,2,3} for any prime p. The research method is determining all idempotent elements and regular elements in T3(ℤp2), (ii) constructing all the strongly r-clean elements. The result showed it can be constructed four general forms of idempotent elements and eight general forms of regular elements in T3(ℤp2). Based on the existence of idempotent and regular elements, it can be constructed twenty general forms of strongly r-clean elements in T3(ℤp2 ) for any prime p.