Generalization of Strongly Clean Rings

TL;DR

This paper extends the concept of strongly clean rings to n-strongly clean and { extSigma}-strongly clean rings, providing examples, structural properties, and characterizations, especially in the context of commutative rings and polynomial rings.

Contribution

It introduces generalized forms of strongly clean rings, explores their properties, and establishes key equivalences and non-existence results in polynomial rings.

Findings

Existence of rings that are n-strongly clean and { extSigma}-strongly clean but not strongly clean.

R[(x)] is n-strongly clean if and only if R is n-strongly clean for commutative rings.

Polynomial rings R[x] are not { extSigma}-strongly clean for any commutative ring R.

Abstract

In this paper, strongly clean ring defined by W. K. Nicholson in 1999 has been generalized to n-strongly clean, {\Sigma}-strongly clean and with the help of example it has been shown that there exists a ring, which is n-strongly clean and {\Sigma}-strongly clean but not strongly clean. It has been shown that for a commutative ring R formal power series R[(x)] of R is n-strongly clean if and only if R is n- strongly clean. We also discussed the structure of homomorphic image of n- strongly clean and direct product of n- strongly clean rings. It has also been shown that for any commutative ring R, the polynomial ring R (x) is not {\Sigma}-strongly clean ring.