(untitled) (Communications in Mathematical Research, 2021)
A ring R is said to be weakly semicommutative if for any a, b ∈ R, ab = 0 implies aRb ⊆ Nil(R), where Nil(R) is the set of all nilpotent elements in R. In this note, we clarify the relationship between weakly semicommutative rings and NI-rings by proving that the notion of a weakly semicommutative ring is a proper generalization of NI-rings. We say that a ring R is weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical, and prove that if R is a weakly 2-primal ring which satisfies α-condition for an endomorphism α of R (that is, ab = 0 ⇔ aα(b) = 0 where a, b ∈ R) then the skew polynomial ring R[x; α] is a weakly 2-primal ring, and that if R is a ring and I is an ideal of R such that I and R/I are both weakly semicommutative then R is weakly semicommutative. Those extend the main results of Liang et al. 2007 (Taiwanese J. Math., 11(5)(2007), 1359–1368) considerably. Moreover, several new results about weakly semicommutative rings and NI-rings are included.