Симметризация в чистых и ниль-чистых кольцах

Peter V Danchev

doi:10.18500/1816-9791-2020-20-2-154-160

Abstract

Симметризация в чистых и ниль-чистых кольцах

Highlights

It is pretty easy to check that nil-clean rings are always clean, but not the converse
The condition e ∈ aRa is obviously equivalent to e ∈ aR ∩ Ra as aRa ⊆ aR ∩ Ra and if e = ab = ca for some b, c ∈ R it follows that e = e.e = abca ∈ aRa as needed. This illustrates that D-nil-clean rings are both L-nil-clean and R-nil-clean; the converse is still unknown
Since 2 ∈ N il(R) by simple operations — omitting some details — we find that 2e + q ∈ N il(R) and 2e + q − 1 ∈ U (R)

Summary

Introduction and Background

Throughout this paper, all rings R are assumed to be associative and unital with identity element 1 different from the zero element 0 of R. The condition e ∈ aRa is obviously equivalent to e ∈ aR ∩ Ra as aRa ⊆ aR ∩ Ra and if e = ab = ca for some b, c ∈ R it follows that e = e.e = abca ∈ aRa as needed This illustrates that D-nil-clean rings are both L-nil-clean and R-nil-clean; the converse is still unknown. In [6] we defined the two concepts of regularly nil clean rings and Utumi rings as follows: a ring R is regularly nil clean if, for every a ∈ R, there is e ∈ Ra∩Id(R) such that a(1 − e) ∈ N il(R) and (1 − e)a ∈ N il(R) or, in an equivalent form, there is f ∈ aR ∩ Id(R) such that a(1 − f ) ∈ N il(R) and (1 − f )a ∈ N il(R). The work is structured as follows: the section states and proves our major results (see, respectively, Propositions 1, 2 and 3, Lemma 1, as well as Remark 1 listed below): the final part consists of some useful commentaries on the more insightful exploration of the current subject and a list of problems that remain open

Preliminary and Main Results

Concluding Discussion and Open Questions