Commutativity of rings satisfying certain polynomial identities

Some Polynomial Identities that Imply Commutativity of Rings

In this paper, we establish some commutativity theorems for certain rings with polynomial constraints as follows: Let R be an associative ring, and for all x, y ∈ R, and fixed non-negative integers m > 1, n ≥ 0, r > 0, s ≥ 0, t ≥ 0, p ≥ 0, q ≥ 0 such that P (x, y) = ±Q(x, y), where P (x, y) = ys[x, y]yt and Q(x, y) = xp[xm, yn]ryq. First,it is shown that a semiprime ring R is commutative if and...

Commutativity for a Certain Class of Rings

We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr[xs, y] = ±[x, yt]xn, xr[xs, y] = ±xn[x, yt], xr[xs, y] = ±[x, yt]ym, and xr[xs, y] = ±ym[x, yt], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we ...

How much commutativity is needed to prove polynomial identities?

Let f be a non-commutative polynomial such that f = 0 if we assume that the variables in f commute. Let Q(f) be the smallest k such that there exist polynomials g1, g ′ 1, g2, g ′ 2, . . . , gk, g ′ k with f ∈ I([g1, g 1], [g2, g 2], . . . , [gk, g k]) , where [g, h] = gh − hg. Then Q(f) ≤ ` n 2 ́ , where n is the number of variables of f . We show that there exists a polynomial f with Q(f) = Ω(...

On Derandomizing Tests for Certain Polynomial Identities

We extract a paradigm for derandomizing tests for polynomial identities from the recent AKS primality testing algorithm. We then discuss its possible application to other tests.

Differential Polynomial Rings of Triangular Matrix Rings