Open AccessОдно замечание о периодических кольцах
Author(s) -
Peter V. Danchev
Publication year - 2021
Publication title -
vladikavkazskij matematičeskij žurnal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.126
H-Index - 2
eISSN - 1814-0807
pISSN - 1683-3414
DOI - 10.46698/q0369-3594-2531-z
Subject(s) - nilpotent , element (criminal law) , mathematics , characterization (materials science) , locally nilpotent , ring (chemistry) , property (philosophy) , state (computer science) , algebra over a field , pure mathematics , combinatorics , discrete mathematics , physics , nilpotent group , philosophy , chemistry , organic chemistry , epistemology , algorithm , political science , law , optics
We obtain a new and non-trivial characterization ofperiodic rings (that are those rings $R$ for which, for eachelement $x$ in $R$, there exists two different integers $m$, $n$strictly greater than $1$ with the property $x^m=x^n$) in terms ofnilpotent elements which supplies recent results in this subject byCui--Danchev published in (J. Algebra \& Appl., 2020) and byAbyzov--Tapkin published in (J. Algebra \& Appl., 2022). Concretely,we state and prove the slightly surprising fact that an arbitraryring $R$ is periodic if, and only if, for every element~$x$from $R$, there are integers $m>1$ and $n>1$ with $m\not= n$ suchthat the difference $x^m-x^n$ is a nilpotent.