Röhmel's equation for quadratic forms (Q1095179)

Let \(q: x\to A\) satisfy \(q(0)=0\) and \(q(ax+y)+q(x-ay)=(1+a)[q(x)+q(y)]\) for all x,y\(\in X\) and \(a\in A\), where A is a commutative ring with 1 and X is an A-module. The author proves that q is a quasi-quadratic form if the only \(k\in A\), which satisfies \(2k=0\) and \((a^ 4-a^ 2)k=0\) for all a in A, is \(k=0\). It is also proved that a quasi-quadratic form becomes a quadratic form if and only if the only function \(D: A\to A\) satisfying \(D(a+b)=D(a)+D(b),\) \(D(1)=0\) and \(D(a^ 2b)=a^ 2 D(b)\) is \(D=0\).