A condition for graded freeness (Q1381655)

We consider a graded associative algebra \(A=\bigotimes_{i\geq 0}A_i\) over a field \(k=A_0\). If \(A\) is a graded free algebra, then any one-sided graded ideal of \(A\) is free -- this fact immediately follows from Shirshov's lemma or from the well-known theorem of P. M. Cohn. In the present article, all graded \(A\)-modules are supposed to be bounded below, i.e., \(M_n=0\) for \(n\ll 0\). We denote by \(k\) the \(A\)-bimodule \(A/A_+\). The inversion of the proposition mentioned is the Theorem. If \((A_+)_A:=A=\bigotimes_{i>0}A_i\) is a graded free \(A\)-module, then \(A\) is a graded free \(k\)-algebra.