Generalization of Strongly Clean Rings

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.