Narrow Results

Let S be a semigroup. The degree of S is the smallest natural number r such that for each x ∈ S, x^n(x)+r = x^n(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.