On the relation between one-sided duoness and commutators - Kim Nam Kyun; Lee Yang (Open Mathematics)
This article studies the structure of rings RR over which the 2×22× 2 upper triangular matrix rings with the same diagonal are right duo, denoted by D2(R)D_{2}\left(R). We prove that for any right regular element dd of such a ring RR, dRdR contains the ideal of RR generated by all commutators. It is also proved that for a domain RR, D2(R)D_{2}\left(R) is right duo if and only if RR is either commutative or a division ring. Moreover, it is proved that if RR is a ring of characteristic 2 such that D2(R)D_{2}\left(R) is right duo, then RR has an ascending chain of nil ideals NRi⊆NRi+1N_{R_{i}}\subseteq {N}_{R_{i+1}} (i=0,1,…i=0,1,… ) such that NRi+1⁄NRiN_{R_{i+1}}/N_{R_{i}} is contained in the center of R⁄NRiR/N_{R_{i}}. Furthermore, we give a simpler proof to the famous result that if RR is a simple noncommutative ring then RR coincides with its subring generated by all commutators (by Herstein). Finally, we show that if D2(R)D_{2}\left(R) is right duo over a ring RR, then Sa⊆aSSa\subseteq aS for any a∈Ra\in R, where SS is any of the following: (i) the prime radical, (ii) the Jacobson radical, (iii) the group of all units in RR, and (iv) the set of one-sided zero-divisors.