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We introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concepts of McCoy rings and nil-Armendariz rings. It is proved that every semicommutative ring is nil-McCoy. We shall give an example to show that nil-McCoy rings need not be semicommutative. Moreover, we show that nil-McCoy rings need not be right linearly McCoy. More examples of nil-McCoy rings are given by various extensions. On the other hand, the properties of α-McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x] are also investigated. For a monomorphism α of a ring R, it is shown that if R is weak α-rigid and α-reversible then R is α-McCoy.
We prove that every nilpotent group of class 2 and exponent 4 is the circle group of a nilpotent ring of index 3 and characteristic 2.
Abstract
It is shown that if N(R) is a Lie ideal of R (respectively Jordan ideal and R is 2-torsion-free), then N(R) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).
A ring R is called NLI (rings whose nilpotent elements form a Lie ideal) if for each a ∈ N(R) and b ∈ R, ab − ba ∈ N(R). Clearly, NI rings are NLI. In this note, many properties of NLI rings are studied. The main results we obtain are the following: (1) NLI rings are directly finite and left min-abel; (2) If R is a NLI ring, then (a) R is a strongly regular ring if and only if R is a Von Neumann regular ring; (b) R is (weakly) exchange if and only if R is (weakly) clean; (c) R is a reduced ring if and only if R is a n-regular ring; (3) If R is a NLI left MC2 ring whose singular simple left modules are Wnil-injective, then R is reduced.
Abstract
Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x n=x n +1 y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x m=x n. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained.
