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We look at division rings in the variety of strongly regular rings and show a connection to the study of rational identities on division rings.
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.
We obtain asymptotic formulas with arbitrary order of accuracy for the eigenvalues and eigenfunctions of a nonselfadjoint ordinary differential operator of order n whose coefficients are Lebesgue integrable on [0, 1] and the boundary conditions are strongly regular. The orders of asymptotic formulas are independent of smoothness of the coefficients.
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.
the different classes. We focus on Frisian regular verbs, and not yet consider any irregular verbs in this paper. 1 Interestingly, we show that the same technique can be applied to both weak and strong regular verbs. As an example of the difference in
