Some new characterizations of periodic rings

A ring [Formula: see text] is called periodic if, for every [Formula: see text] in [Formula: see text], there exist two distinct positive integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. The paper is devoted to a comprehensive study of the periodicity of arbitrary unital rings. Some new characterizations of periodic rings and their relationship with strongly [Formula: see text]-regular rings are provided as well as, furthermore, an application of the obtained main results to a ∗-version of a periodic ring is being considered. Our theorems somewhat considerably improved on classical results in this direction.

A Note on Periodic Rings

Let S be a semigroup. The degree of S is the smallest natural number r such that for each x ∈ S, xn(x)+r = xn(x), where n(x) ∈ ℕ. If such a number r does not exist, we say that the degree of S is infinite. For a group G, this coincides with the exponent of G. We prove that for a periodic ring R, the degree of R equals exp (U(R)), where U(R) denotes the unit group of R. Then we determine all degrees for any rings.