An analogue of Hilbert's syzygy theorem for the algebra of one-sided inverses of a polynomial algebra. (Q2845850)

Let \(A\) be an associative ring with unit element. Denote by \(S_n(A)\) the ring generated over \(A\) by elements \(x_1,\ldots,x_n,y_1,\ldots,y_n\) with defining relations \(y_ix_i=1\), \([x_i,x_j]=[x_i,y_j]=[y_i,y_j]=0\). The ring \(S_n(A)\) is obtained from the polynomial ring \(A[x_1,\ldots,y_n]\) by adding left commuting inverses of the variables \(x_1,\ldots,x_n\). It is shown that the left (right) global dimension of \(S_n(A)\) is equal to the left (right) global dimension of \(A\) plus \(n\).