The algebra of one-sided inverses of a polynomial algebra

We study in detail the algebra Sn in the title which is an algebra obtained from a polynomial algebra Pn in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of Pn. The algebra Sn is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of Sn is 2n; but the weak and the global dimensions of Sn are n. The prime and maximal spectra of Sn are found, and the simple Sn-modules are classified. It is proved that the algebra Sn is central, prime, and catenary. The set In of idempotent ideals of Sn is found explicitly. The set In is a finite distributive lattice and the number of elements in the set In is equal to the Dedekind number dn.