Commutative rings with homomorphic power functions (Q1186335)

A commutative ring \(R\) with identity is said to be \(m\)-linear (for an integer \(m\geq 2)\) if \((a+b)^ m=a^ m+b^ m\) for all \(a,b\in R\). This paper gives a careful study of \(m\)-linear rings. The authors show that a ring \(R\) is \(m\)-linear if and only if \(R\) is a finite product of \(m\)- linear rings of prime characteristic. The main result is then that for each prime \(p\) and integer \(m\geq 2\) which is not a power of \(p\), there exists an integer \(s\geq m\) such that, for each ring \(R\) of characteristic \(p\), \(R\) is \(m\)-linear if and only if \(r^ m=r^{p^ s}\) for each \(r\in R\). Many other results and examples are given.