Nil-clean matrix rings

Abstract

We characterize the nil clean matrix rings over fields. As a by product, we obtain a complete characterization of the finite rank Abelian groups with nil clean endomorphism ring and the Abelian groups with strongly nil clean endomorphism ring, respectively. * S. Breaz is supported by the CNCS-UEFISCDI grant PN-II-RU-TE-2011-3-0065. †

Nil-clean and strongly nil-clean rings

“…Here, we work on this question focusing on some classes of rings for which the Köthe conjecture holds true. Let us recall some known progress: For a field or a division ring D, M n (D) is nil-clean iff D ∼ = F 2 (see [5,16]); for a strongly regular ring R, M n (R) is nil-clean iff R is boolean (see [16]). In this section, we will provide some further improving answers to the aforementioned question.…”

“…Furthermore, the nil-cleanness of a matrix ring is tightly linked to the famous Köthe Conjecture (see Section 6). The reader is referred to the papers [10,5,16,2] for the background and current stage of the study of nil-clean and strongly nil-clean rings.…”

Weakly Nil-Clean Index and Uniquely Weakly Nil-Clean Rings

“…When studying weakly nil-clean matrices, it is not enough to study companion matrices which are (or are not) blocks of other companion matrices. In fact, note that not all matrices are similar to a companion matrix (see the proof of the main result in [2] or [4]). …”

ON WEAKLY $g(x)$-NIL CLEAN RINGS

“…In [4], Breaz, Danchev and Zhou defined a ring R to be weakly nil clean if each element r ∈ R can be written as r = n + e or r = n − e for n ∈ N (R) and e ∈ Id(R). We refer the reader to [3,7,8,10,19,4] for a survey on nil clean and weakly nil clean rings.…”