The global dimension of the algebras of polynomial integro-differential operators In and the Jacobian algebras An

The aim of the paper is to prove two conjectures that the (left and right) global dimension of the algebra of polynomial integro-differential operators In and the Jacobian algebra An is equal to n (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras In and An are neither left nor right Noetherian. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. It is proven that the global dimension of all prime factor algebras of the algebras In and An is n and the weak global dimension of all the factor algebras of In and In is n

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Qcl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Ql(R) of an arbitrary ring R is proved, i.e., Ql(R) = S0(R)−1R where S0(R) is the largest left regular denominator set of R. It is proved that Ql(Ql(R)) = Ql(R); the ring Ql(R) is semisimple iff Qcl(R) exists and is semisimple; moreover, if the ring Ql(R) is left Artinian, then Qcl(R) exists and Ql(R) = Qcl(R). The group of units Ql(R)* of Ql(R) is equal to the set {s−1t | s, t ∈ S0(R)} and S0(R) = R ∩ Ql(R)*. If there exists a finitely generated flat left R-module which is not projective, then Ql(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators)

The Jacobian algebras

A new class of algebras (the Jacobian algebras) is introduced and studied in detail. The Jacobian algebras are obtained from the Weyl algebras by inverting (not in the sense of Ore) certain elements. Surprisingly, the Jacobian algebras and the Weyl algebras have little in common. Moreover, they have almost opposite properties

Classification of simple modules of the ore extension K[X][Y;fddX]

For the algebras Λ in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of Λ are computed

Criteria for a Ring to have aLeft Noetherian Largest Left Quotient Ring

Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets

The groups of automorphisms of the Lie algebras of polynomial vector fields with zero or constant divergence

Let Pn = K[x1, . . . , xn] be a polynomial algebra over a eld K of characteristic zero and div0 n (respectively, divc n ) be the Lie algebra of derivations of Pn with zero (respectively, constant) divergence. We prove that AutLie(div0 n ) ≃ AutK−alg(Pn) (n ≥ 2) and AutLie(divc n ) ≃ AutK−alg(Pn). The Lie algebra divc n is a maximal Lie subalgebra of DerK (Pn). Minimal nite sets of generators are found for the Lie algebras div0 n and divc n

Skew category algebras

We study a new (large) class of algebras (that was introduced in Bavula in Math Comput Sci 11(3–4):253–268, 2017)—the skew category algebras. Any such an algebra C(σ) is constructed from a category C and a functor σ from the category C to the category of algebras. Criteria are given for the algebra C(σ) to be simple or left Noetherian or right Noetherian or semiprime or have 1