\(\mathbb Z\)-graded simple rings. (Q2790628)

Let \(k\) be an algebraically closed field and \(A=\bigoplus _{i\in\mathbb Z}A_i\) a graded simple affine associative \(k\)-algebra. Suppose that \(\mathrm{GK-}\dim A=2\) and each \(A_i\neq 0\). Then in the first case the graded quotient ring of \(A\) has the form \(K[t^{\pm 1};\sigma]\) where \(K=k(u)\) is transcendental extension of \(k\) of degree 1 and one can choose \(u\) of the form either \(\sigma(u)=u+1\) where \(A_0=k[u]\), \(\mathrm{char }k=0\), or \(\sigma(u)=pu\), \(p\in k^*\). In the second case \(A\) is graded Morita equivalent to generalized Weyl algebra in the sense of V. Bavula. This result under some additional assumptions is extended where graded quotient ring of \(A\) has the same form in which \(K\) has transcendence degree \(d\) and \(\mathrm{GK-}\dim A=d+1\). The case of Morita equivalence in the first result is considered in details.