The algebra of one-sided inverses of a polynomial algebra

الأصل · EN

We study in detail the %Shrek algebra ₙ 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 ₙ 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 ₙ is 2n; but the weak and the global dimensions of ₙ are n. The prime and maximal spectra of ₙ are found, and the simple ₙ-modules are classified. It is proved that the algebra ₙ is central, prime, and catenary. The set ₙ of idempotent ideals of ₙ is found explicitly. The set ₙ is a finite distributive lattice and the number of elements in the set ₙ is equal to the Dedekind number ₙ.