A ring means an associative ring. Let \(R\) be a ring, let \(S\) be a semigroup, and let \(S^*\) be the set of all nonzero elements of \(S\). Suppose \(\sigma:S^*\to\text{End }R\) is a mapping satisfying the condition: if \(a,b,ab\in S^*\) then \(\sigma(ab)=\sigma(a)\sigma(b)\). Using \(\sigma\), the authors define a skew semigroup ring of \(S\) over \(R\), the concept analogous to the concept of a skew polynomial ring. The authors investigate some properties of skew semigroup rings. In particular, they consider the problem when these rings are finite unital normalizing extensions of \(R\) (a unital ring \(B\) is a finite normalizing extension of its unital subring \(A\) iff there exists a finite set \(\{b_i\}\subset B\) such that \(B=\sum Ab_i\), and \(Ab_i=b_iA\) for all \(b_i\)). The dual concept in which the claim \(\sigma(ab)=\sigma(a)\cdot\sigma(b)\) is changed to the claim \(\sigma(ab)=\sigma(b)\sigma(a)\) is considered also.