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On strongly nil clean rings

机译:On strongly nil clean rings

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A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a is an element of R, there exists an idempotent e is an element of Za such that a - e is an element of N(R), if and only if R is periodic and R/J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove hat R is strongly nil clean if and only if for all a is an element of R, there exist n is an element of N, k >= 0 depending on a) such that a(n) - a(n+2k) is an element of N(R), if and only if for fixed m, n is an element of N, a - a(2n+2m(n+1)) is an element of N(R) for all a is an element of R. These also extend known theorems, e.g, 5, Theorem 3.21, 6, Theorem 3, 7, Theorem 2.7 and 12, Theorem 2.

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