A theorem of T. Nakayama states that an algebra \(A\) over an \({\mathcal N}\)- ring \(R\) is commutative if \(A\) satisfies the following condition: (N) For each \(x\) in \(A\), there exists \(f(X)\) in \(X^ 2 R[X]\) such that \(x-f(x)\) is central. More generally, W. Streb studied \(R\)-algebras \(A\) satisfying the following condition: (S) For each \(x\), \(y\) in \(A\), there exists \(f(X,Y)\) in \(K_ 3\) such that \([x,y]=f(x,y)\). Recently, he gave a classification of non-commutative rings. Y. Kobayashi characterized a certain class of rings with the identity \([X^ n,Y^ n]=0\). First we give two theorems extending the results of Streb to algebras, and characterize the class of algebras with (S) and the identity \([X^ n,Y^ n]=0\).