Modules over the algebra of the noncommutative equation \(yx=1\)
The \(K\)-algebra \(A\) generated by elements \(x\) and \(y\) with defining relation \(yx=1\) has already been studied extensively in previous work by Jacobson. In this paper, the author presents a structure theorem for the category \(\text{mod-}A\) of right \(A\)-modules. His main result states that this category may essentially be identified with that of extensions \(0\to M'\to M\to\overline M\to 0\), where \(\overline M\) is an \(A\)-module annihilated by the two-sided ideal \(S\) generated by \([y,x]\) and \(M'S=M'\). The author also presents an explicit description of the right ideals of \(A\) as well as the right Rosenberg spectrum \(\text{Spec}_r(A)\). In particular, it thus appears that \(A\) may be viewed as the algebra of a non-commutative torus. Finally, it is shown that one can construct a non-commutative projective line by adjoining points at infinity; this may be done by constructing an Abelian category of coherent modules relative to a covering by a pair of affine lines.
scientific article; zbMATH DE number 1538409