Open AccessCharacterizations of special clean elements and applications
Author(s) -
Xavier Mary,
Pedro Patrı́cio
Publication year - 2020
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil2014847m
Subject(s) - mathematics , constructive , mathematical proof , uniqueness , simple (philosophy) , element (criminal law) , ring (chemistry) , set (abstract data type) , algebra over a field , pure mathematics , calculus (dental) , discrete mathematics , mathematical analysis , geometry , computer science , law , medicine , philosophy , chemistry , organic chemistry , process (computing) , epistemology , dentistry , political science , programming language , operating system
We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring. We then derive ?constructive? proofs that in many rings, regular elements are special clean by solving this equation in specific cases. Other applications, such as uniqueness of decompositions, are given. Many examples of special clean decompositions of 2-2 matrices found by this methodology are also presented.