Lifting potent elements modulo nil ideals (Q2031547)

An element \(x\) of a ring \(R\) (always with identity; not necessarily commutative) is called potent if \(x^{n}=x\) for some integer \(n>1\) . The smallest such \(n\) is the index of potency. An idempotent element is a potent element of potency two. Any potent element is periodic. Let \(I\) be a one-sided ideal of a ring \(R\). An element \(x\in R\) is said to be \textit{ potent modulo} \(I\) if \(x^{n}-x\in I\) for a positive integer \(n>1\). A potent element \(x\) modulo \(I\) \textit{lifts} if there is a potent element \(p\in R\), possibly of different index of potency, such that \(p-x\in I\). Similarly, a torsion unit \(x\) modulo \(I\) lifts if there exists a torsion unit \(t\in R\), possibly of different order, such that \(t-x\in I\). Lifting of various kinds of elements is not a new problem and many investigations in this regard have been conducted. It is known that idempotents lift modulo nil one-sided ideals and it raises the question on the validity of this statement for potent elements. Here the authors investigate the lifting problem for potent and other types of ring elements. It is shown that for a nil ideal \(I\) of an abelian ring \(R\), torsion units lift modulo \(I\) precisely when potent elements or periodic elements lift modulo \(I\). An example is given to show that torsion units may not lift modulo a nilpotent ideal \(I\) of index two in a commutative ring \(R\). It follows that potent elements and periodic elements also may not lift modulo nilpotent ideals of index two in commutative rings and so potent elements do not lift modulo nil one-sided ideals in general. Several special cases when it does hold is given. In particular, it is shown that torsion units, potent elements and periodic elements lift modulo nil one-sided ideals of a ring with finite characteristic; torsion units and potent elements lift modulo every nil one-sided ideal; and periodic elements lift modulo every nil ideal of a \(\pi \)-regular ring. Moreover, torsion units, potent elements and periodic elements lift modulo every ideal of a local ring with nil Jacobson radical. A generalization that idempotents lift modulo nil one sided ideals is also given: If \(I\) is a nil one sided ideal of \(R\) and \(x\in R\) is such that \(x^{n+1}-x\in I\) for some positive integer \(n\) and \(n\) is a unit in \(R\) , then there exists an element \(p\in R\) with \(p^{n+1}=p\) such that \(x-p\in I\) .