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The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on commutative algebras (in arbitrary characteristic), all subalgebras of $\CD (A)$ that contain the algebra $A$, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions $A[x_1, \ldots , x_n ; \d_1 , \ldots , \d_n]$, certain generalized Weyl algebras, and others. In \cite{SimCrit-difop}, simplicity criteria are given for the algebras differential operators on commutative algebras (it was a long standing problem). The aim of the paper is to describe the ideal structure of $\D$-locally nilpotent algebras and as a corollary to give simplicity criteria for them (it is a generalization of the results of \cite{SimCrit-difop}). Examples are considered., Comment: 19 pages

The aim of the papers is to describe the left regular left quotient ring ${}'Q(R)$ and the right regular right quotient ring $Q'(R)$ for the following algebras $R$: $\mS_n=\mS_1^{\t n}$ is the algebra of one-sided inverses, where $\mS_1=K\langle x,y\, | \, yx=1\rangle$, $\CI_n=K\langle \der_1, \ldots, \der_n,\int_1,\ldots, \int_n\rangle$ is the algebra of scalar integro-differential operators and the Jacobian algebra $\mA_1=K\langle x,\der, (\der x)^{-1}\rangle$. The sets of left and right regular elements of the algebras $\mS_1$, $\CI_1$, $\mA_1$ and $\mI_1=K\langle x, \der,\int\rangle$. A progress is made on the following conjecture, \cite{Clas-lreg-quot}: $${}'Q(\mI_n)\simeq Q(A_n)\;\; {\rm where}\;\; \mI_n =K\bigg\langle x_1,\ldots , x_n, \der_1, \ldots, \der_n,\int_1,\ldots, \int_n\bigg\rangle$$ is the algebra of polynomial integro-differential operators and $Q(A_n)$ is the classical quotient ring (of fractions) of the $n$'th Weyl algebra $A_n$, i.e. a criterion is given when the isomorphism holds. We produce several general constructions of left Ore and left denominator sets that appear naturally in applications and are of independent interest and use them to produce explicit left denominator sets that give the localization ring isomorphic to ${}'Q(\mS_n)$ or ${}'Q(\mI_n)$ or ${}'Q(\mA_n)$ where $\mA_n:=\mA_1^{\t n}$. Several characterizations of one-sided regular elements of a ring are given in module-theoretic and one-sided-ideal-theoretic way., Comment: 32 pages

In \cite[Section 5, p.32]{Arnold-1998}, Arnold writes: "Classification of singularities of curves can be interpreted in dual terms as a description of 'co-artin' subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)." In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let $K$ be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras $\mathcal{A}$ of the polynomial algebra $K[x]$ that contains the ideal $x^mK[x]$ for some $m\geq 1$. It is proven that the set $\mathcal{A} = \coprod_{m, \Gamma }\mathcal{A} (m, \Gamma )$ is a disjoint union of affine algebraic varieties (where $\Gamma \coprod \{0, m, m+1, \ldots \}$ is the semigroup of the singularity and $m-1$ is the Frobenius number). It is proven that each set $\mathcal{A} (m, \Gamma )$ is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on $\mathcal{A} (m ,\Gamma )$. An isomorphism criterion is given for the algebras in $\mathcal{A}$. For each algebra $A\in \mathcal{A} (m, \Gamma)$, explicit sets of generators and defining relations are given and the automorphism group ${\rm Aut}_K(A)$ is explicitly described. The automorphism group of the algebra $A$ is finite iff the algebra $A$ is not isomorphic to a monomial algebra, and in this case $|{\rm Aut}_K(A)|<{\rm dim}_K(A/\mathfrak{c}_A)$ where $\mathfrak{c}_A$ is the conductor of $A$. The set of orders of the automorphism groups of the algebras in $\mathcal{A} (m , \Gamma )$ is explicitly described., Comment: 29 pages

The concepts of localizable set, localization of a ring and a module at a localizable set are introduced and studied. Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a localizable set is a generalization of localization of a ring/module at a denominator set. For a semiprime left Goldie ring, it is proven that the set of maximal left localizable sets that contain all regular elements is equal to the set of maximal left denominator sets (and they are explicitly described). For a semiprime Goldie ring, it is proven that the following five sets coincide: the maximal Ore sets, the maximal denominator sets, the maximal left or right or two-sided localizable sets that contain all regular elements (and they are explicitly described)., Comment: 23 pages

The aim of the paper is to give an explicit description of bi-quadratic algebras on 3 generators with PBW basis., Comment: 24 pages

A polynomial endomorphism $\sigma\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $\sigma$ is extended to an endomorphism $\sigma$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $\sigma$) $P_n$-module ${}^{\sigma} P_n$ is 1-generated for all Jacobian maps $\sigma$. It is shown that the $A_n$-module ${}^{\sigma} P_n$ is 1-generated for all Jacobian maps $\sigma$. Furthermore, the $A_n$-module ${}^{\sigma} P_n$ is holonomic and as a result has finite length. An explicit upper bound is found for the length of the $A_n$-module ${}^{\sigma} P_n$ in terms of the degree ${\rm deg} (\sigma )$ of the Jacobian map $\sigma$. Analogous results are given for the Conjecture of Dixmier and the Poisson Conjecture. These results show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$, the images of the Jacobian maps, endomorphisms of the Weyl algebra $A_n$ and the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would be either to prove the conjectures or produce counter examples. A short direct algebraic (without reduction to prime characteristic) proof is given of equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of equivalence of the Jacobian, Poisson and Dixmier Conjectures)., Comment: 7 pages

Let $\Lambda (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over an arbitrary field $K$ of characteristic $p>0$ where $f\in K[x]$. For each polynomial $f$, the automorphism group of the algebras $\Lambda (f)$ is explicitly described. The automorphism group ${\rm Aut}_K(\L (f))=S\rtimes G_f$ is a semidirect product of two explicit groups where $G_f$ is the {\em eigengroup} of the polynomial $f$ (the set of all automorphisms of $K[x]$ such that $f$ is their common eigenvector). For each polynomial $f$, the eigengroup $G_f$ is explicitly described. It is proven that every subgroup of ${\rm Aut}_K(K[x])$ is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra $\Lambda (f)$ are 2. The prime, completely prime, primitive and maximal ideals of the algebra $\Lambda (f)$ are classified., Comment: 31 pages

Let $\L (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For a given polynomial $f$, the automorphism group of the algebra $\L (f) $ is explicitly described. The polynomial case $\L (0) = K[x,y]$ and the case of the Weyl algebra $A_1= K[x][y; \frac{d}{dx} ]$ were done done by Jung (1942) and van der Kulk (1953), and Dixmier (1968), respectively. In 1997, Alev and Dumas proved that the algebras $\L (f)$ and $\L (g)$ are isomorphic iff $g(x) = \l f(\alpha x+\beta )$ for some $\l, \alpha \in K\backslash \{ 0\}$ and $\beta\in K$. In 2015, Benkart, Lopes and Ondrus gave a complete description of the set of automorphism groups of algebras $\L(f)$. In this paper we complete the picture, i.e. {\em given} the polynomial $f$ we have the explicit description of the automorphism group of $\L (f)$., Comment: 11 pages

For a generalized Weyl Poisson algebra $A$, explicit sets of generators and defining relations are presented for its Poisson enveloping algebra $\CU (A)$. Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poisson differential operators $P\CD (A)$ on $A$. The Gelfand-Kirillov dimensions of the algebras $\CU (A)$ and $P\CD (A)$ are calculated. It is proven that the algebra $\CU (A)$ is a domain provided that the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is a domain of essentially finite type over a perfect field. For the algebra $A$, the set of its minimal primes and the prime radical are described and an equidimensionality criterion is given. For the equidimensional algebra $A$ of essentially finite type, two regularity criteria are presented., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:2107.00321

For an arbitrary Poisson algebra $\CP$ over an arbitrary field, an (analogue of) the Poincar\'{e}-Birkhof-Witt Theorem is proven and several presentations/constructions for its Poisson enveloping algebra $\CU (\CP )$ are given. As a result, explicit sets of generators and defining relations are given for $\CU (\CP )$ and the algebra $P\CD (\CP)$ of Poisson differential operators on $\CP$. Simplicity criteria for the algebras $\CU (\CP )$ and $P\CD (\CP )$ are given. In the case when the algebra $\CP$ is of essentially finite type, a criterion for the algebra $\CU (\CP )$ to be a domain is presented and a criterion for a natural epimorphism $\CU (\CP )\ra P\CD (\CP )$ to be an isomorphism is given. The kernel of the epimorphism is described and for large classes of Poisson algebras an explicit set of generators is given. Explicit formulae for the Gelfand-Kirillov dimension of the algebras $\CU (\CP )$ and $P\CD (\CP)$ are given. In the case when the Poisson algebra $\CP$ is a regular domain of essentially finite type an explicit simplecticity criterion for $\CP$ is found and a criterion is presented for the algebra $\CU (\CP )$ to be isomorphic to the algebra $\CD (\CP )$ of differential operators on $\CP$., Comment: 41 pages

Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal}, $\CD (\CA )$ be the algebra of differential operators on the algebra $\CA$. The aim of the paper is to give a simplicity criterion for the algebra $\CD (\CA )$: {\em The algebra $\CD (\CA )$ is simple iff $\CD (\CA ) \ga_r^i\CD (\CA )= \CD (\CA )$ for all $i\geq 1$ provided the field $K$ is a perfect field.} Furthermore, a simplicity criterion is given for the algebra $\CD (R)$ of differential operators on an arbitrary commutative algebra $R$ over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators., Comment: 5 pages

For the algebra $I_n$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $I_n$-modules is given. It is proven that the category of weight $I_n$-modules is semisimple. An explicit description of generalized weight $I_n$-modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $I_n$-modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $I_n$-modules is given. In the wild case `natural` tame subcategories are considered with explicit description of indecomposable modules. It is proven that every generalized weight $I_n$-module is a unique sum of absolutely prime modules. For an arbitrary ring $R$, we introduce the concept of {\em absolutely prime} $R$-module (a nonzero $R$-module $M$ is absolutely prime if all nonzero subfactors of $M$ have the same annihilator). It is shown that every indecomposable generalized weight $I_n$-module is equidimensional. A criterion is given for a generalized weight $I_n$-module to be finitely generated., Comment: 21 pages

A new class of Poisson algebras, the class of {\em generalized Weyl Poisson algebras}, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl algebras. A Poisson simplicity criterion is given for generalized Weyl Poisson algebras and explicit descriptions of the Poisson centre and the absolute Poisson centre are obtained. Many examples are considered., Comment: 12 pages

The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_1$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P, Q \in A_1$ then $A_1 = K \langle P, Q \rangle$. The Weyl algebra $A_1$ is a $\Z$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A$ (there is no restriction on the total degrees of $P$ and $Q$)., Comment: 7 pages

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 $\mathbb{I}_n$ and the Jacobian algebra $\mathbb{A}_n$ is equal to $n$ (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras $\mathbb{I}_n$ and $\mathbb{A}_n$ 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 $\mathbb{I}_n$ and $\mathbb{A}_n$ is $n$ and the weak global dimension of all the factor algebras of $\mathbb{I}_n$ and $\mathbb{I}_n$ is $n$., Comment: 22 pages. arXiv admin note: text overlap with arXiv:0912.0723

The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old' GWAs. A new class of rings, the {\em diskew polynomial rings}, is introduced that is closely related to GWAs (they are GWAs under a mild condition). The, so-called, ambiskew polynomial rings are a small subclass of the class of diskew polynomial rings. Semisimplicity criteria are given for generalized Weyl algebras and diskew polynomial rings., Comment: 29 pages

For the algebra $A$ in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of $A$ is determined. The simple unfaithful $A$-modules and the simple weight $A$-modules are classified., Comment: 21 pages

Two criteria are given for a ring to have a left Noetherian left quotient ring (this was an open problem since 70's). It is proved that each such ring has only finitely many maximal left denominator sets., Comment: 16 pages

Let $R$ be a ring, $\CC_R$ and $\pCCR$ be the set of regular and left regular elements of $R$ ($\CC_R\subseteq \pCCR$). Goldie's Theorem is a semisimplicity criterion for the classical left quotient ring $Q_{l,cl}(R):=\CC_R^{-1}R$. Semisimplicity criteria are given for the classical left regular left quotient ring $'Q_{l,cl}(R):=\pCCR^{-1}R$. As a corollary, two new semisimplicity criteria for $Q_{l,cl}(R)$ are obtained (in the spirit of Goldie)., Comment: 22 pages

A new class of rings, {\em the class of weakly left localizable rings}, is introduced. A ring $R$ is called {\em weakly left localizable} if each non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1405.4552

A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring has a left Artinian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A characterization is given of the class of rings that are finite direct product of left localization maximal rings., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1303.0859, arXiv:1405.0214

For an arbitrary left Artinian ring $R$, explicit descriptions are given of all the left denominator sets $S$ of $R$ and left localizations $S^{-1}R$ of $R$. It is proved that, up to $R$-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. $S^{-1}R\simeq S_e^{-1}R$ and ${\rm ass} (S) = {\rm ass} (S_e)$ where $S_e=\{1,e\}$ is a left denominator set of $R$ and $e$ is an idempotent. Moreover, the idempotent $e$ is unique up to a conjugation. It is proved that the number of maximal left denominator sets of $R$ is finite and does not exceed the number of isomorphism classes of simple left $R$-modules. The set of maximal left denominator sets of $R$ and the left localization radical of $R$ are described., Comment: 31 pages

The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence are found., Comment: 4 pages

The group of automorphisms is found for the Lie algebra of polynomial vector fields with constant divergence., Comment: 19 pages

For an arbitrary ring $R$, the largest strong left quotient ring $Q_l^s(R)$ of $R$ and the strong left localization radical $\glsR$ are introduced and their properties are studied in detail. In particular, it is proved that $Q_l^s(Q_l^s(R))\simeq Q_l^s(R)$, $\gll^s_{R/\glsR}=0$ and a criterion is given for the ring $ Q_l^s(R)$ to be a semisimple ring. There is a canonical homomorphism from the classical left quotient ring $Q_{l, cl}(R)$ to $Q_l^s(R)$ which is not an isomorphism, in general. The objects $Q_l^s(R)$ and $\gll^s_R$ are explicitly described for several large classes of rings (semiprime left Goldie ring, left Artinian rings, rings with left Artinian left quotient ring, etc)., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1303.0859

We prove that the group of automorphisms of the Lie algebra $\Der_K (Q_n)$ of derivations of the field of rational functions $Q_n=K(x_1,..., x_n)$ over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the $K$-algebra $Q_n$., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1304.3836

The groups of automorphisms of the Witt $W_n$ and Virasoro Lie algebras are found., Comment: 9 pages

We prove that the group of automorphisms of the Lie algebra $\Der_K (P_n)$ of derivations of a polynomial algebra $P_n=K[x_1,..., x_n]$ over a field of characteristic zero is canonically isomorphic to the the group of automorphisms of the polynomial algebra $P_n$., Comment: 9 pages

Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring $R$ the set $\CM$ of maximal left denominator sets of $R$ is a non-empty set: Theorem (The First Criterion). A ring $R$ has a semisimple left quotient ring $Q$ iff $\CM $ is a finite set, $\bigcap_{S\in \CM} \ass (S) =0$ and, for each $S\in \CM$, the ring $S^{-1}R$ is a simple left Artinian ring. In this case, $Q\simeq \prod_{S\in \CM} S^{-1}R$. The Second Criterion is given via the minimal primes of $R$ and goes further then the First one %and Goldie's Theorem in the sense that it describes explicitly the maximal left denominator sets $S$ via the minimal primes of $R$. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets., Comment: 22 pages

Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In this paper, three more new criteria are given., Comment: 16 pages

We prove that every monomorphism of the Lie algebra $\ggu_n$ of unitriangular derivations of the polynomial algebra $P_n=K[x_1,..., x_n]$ is an automorphism., Comment: 4 pages

The groups of automorphisms of the Lie algebras of unitriangular polynomial derivations are found., Comment: 27 pages

The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations., Comment: 33 pages

In contrast to its subalgebra $A_n:=K$ of polynomial differential operators (i.e. the $n$'th Weyl algebra), the algebra $\mI_n:=K$ of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that $\mI_n$ is a left (right) coherent algebra iff $n=1$; the algebra $\mI_n$ is a {\em holonomic $A_n$-bimodule} of length $3^n$ and has multiplicity $3^n$, and all $3^n$ simple factors of $\mI_n$ are pairwise non-isomorphic $A_n$-bimodules. The socle length of the $A_n$-bimodule $\mI_n$ is $n+1$, the socle filtration is found, and the $m$'th term of the socle filtration has length ${n\choose m}2^{n-m}$. This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra $\mI_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra $\mI_n$., Comment: 10 pages

The left quotient ring (i.e. the left classical ring of fractions) $Q_{cl}(R)$ 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 paper, it is proved existence of the largest left quotient ring $Q_l(R)$, i.e. $Q_l(R) = S_0(R)^{-1}R$ where $S_0(R)$ is the largest left regular denominator set of $R$. It is proved that $Q_l(Q_l(R))=Q_l(R)$; the ring $Q_l(R)$ is semi-simple iff $Q_{cl}(R)$ exists and is semi-simple; moreover, if the ring $Q_l(R)$ is left artinian then $Q_{cl}(R)$ exists and $Q_l(R) = Q_{cl}(R)$. The group of units $Q_l(R)^*$ of $Q_l(R)$ is equal to the set $\{s^{-1} t\, | \, s,t\in S_0(R)\}$ and $S_0(R) = R\cap Q_l(R)^*$. If there exists a finitely generated flat left $R$-module which is not projective then $Q_l(R)$ is not a semi-simple 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 (semi-prime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators)., Comment: 32 pages

Let $A_1=K < X, Y | [Y,X]=1>$ be the (first) Weyl algebra over a field $K$ of characteristic zero. It is known that the set of eigenvalues of the inner derivation $\ad (YX)$ of $A_1$ is $\Z$. Let $ A_1\ra A_1$, $X\mapsto x$, $Y\mapsto y$, be a $K$-algebra homomorphism, i.e. $[y,x]=1$. It is proved that the set of eigenvalues of the inner derivation $\ad (yx)$ of the Weyl algebra $A_1$ is $\Z$ and the eigenvector algebra of $\ad (yx)$ is $K< x,y> $ (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: {\em is an algebra endomorphism of $A_1$ an automorphism?})., Comment: 16 pages

Let $A_1:=K\langle x, \frac{d}{dx} \rangle$ be the Weyl algebra and $\mI_1:= K\langle x, \frac{d}{dx}, \int \rangle$ be the algebra of polynomial integro-differential operators over a field $K$ of characteristic zero. The Conjecture/Problem of Dixmier (1968) [still open]: {\em is an algebra endomorphism of the Weyl algebra $A_1$ an automorphism?} The aim of the paper is to prove that {\em each algebra endomorphism of the algebra $\mI_1$ is an automorphism}. Notice that in contrast to the Weyl algebra $A_1$ the algebra $\mI_1$ is a non-simple, non-Noetherian algebra which is not a domain. Moreover, it contains infinite direct sums of nonzero left and right ideals., Comment: 13 pages

For the algebra $\mI_1= K$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple modules is given. It is proved that $\mI_1$ is a left and right coherent algebra. The {\em Strong Compact-Fredholm Alternative} is proved for $\mI_1$. The endomorphism algebra of each simple $\mI_1$-module is a {\em finite dimensional} skew field. In contrast to the first Weyl algebra, the centralizer of a non-scalar integro-differential operator can be a noncommutative, non-Noetherian, non-finitely generated algebra which is not a domain. It is proved that neither left nor right quotient ring of $\mI_1$ exists but there exists the {\em largest left quotient ring} and the {\em largest right quotient ring} of $\mI_1$, they are not $\mI_1$-isomorphic but $\mI_1$-{\em anti-isomorphic}. Moreover, the factor ring of the largest right quotient ring modulo its only proper ideal is isomorphic to the quotient ring of the first Weyl algebra. An analogue of the Theorem of Stafford (for the Weyl algebras) is proved for $\mI_1$: each {\em finitely generated} one-sided ideal of $\mI_1$ is 2-generated., Comment: 47 pages

An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$, Comment: 8 pages

The algebra $\mS_n$ of one-sided inverses of a polynomial algebra $P_n$ in $n$ variables is obtained from $P_n$ by adding commuting, {\em left} (but not two-sided) inverses of the canonical generators of the algebra $P_n$. The algebra $\mS_n$ is a noncommutative, non-Noetherian algebra of classical Krull dimension $2n$ and of global dimension $n$ which is not a domain. If the ground field $K$ has characteristic zero then the algebra $\mS_n$ is canonically isomorphic to the algebra $K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n}, \int_1, ..., \int_n>$ of scalar integro-differential operators. %Ignoring non-Noetherian % property, the algebra $\mS_n$ belongs to a family of algebras % like the $n$th Weyl algebra $A_n$ and the polynomial algebra %$P_{2n}$. It is proved that $ {\rm K}_1(\mS_n)\simeq K^*$. The main idea is to show that the group $\GL_\infty (\mS_n)$ is generated by $K^*$, the group of elementary matrices $E_\infty (\mS_n)$ and $(n-2)2^{n-1}+1$ explicit (tricky) matrices and then to prove that all the matrices are elementary. For each nonzero idempotent prime ideal $\gp$ of height $m$ of the algebra $\mS_n$, it is proved that $$ \rm K}_1(\mS_n, \gp)\simeq K^*, & \text{if}m=1, \Z^{\frac{m(m-1)}{2}}\times K^{*m}& \text{if}m> 1.$$, Comment: 23 pages. arXiv admin note: text overlap with arXiv:0906.3733

The group $\rG_n$ of automorphisms of the algebra $\mI_n:=K< x_1, >..., x_n, \frac{\der}{\der x_1}, ... ,\frac{\der}{\der x_n}, \int_1, >..., \int_n>$ of polynomial integro-differential operators is found: $$ \rG_n=S_n\ltimes \mT^n\ltimes \Inn (\mI_n) \supseteq S_n\ltimes \mT^n \ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}, $$ $$ \rG_1\simeq \mT^1 \ltimes \GL_\infty (K),$$ where $S_n$ is the symmetric group, $\mT^n$ is the $n$-dimensional torus, $\Inn (\mI_n)$ is the group of inner automorphisms of $\mI_n$ (which is huge). It is proved that each automorphism $\s \in \rG_n$ is uniquely determined by the elements $\s (x_i)$'s or $\s (\frac{\der}{\der x_i})$'s or $\s (\int_i)$'s. The stabilizers in $\rG_n$ of all the ideals of $\mI_n$ are found, they are subgroups of {\em finite} index in $\rG_n$. It is shown that the group $\rG_n$ has trivial centre, $\mI_n^{\rG_n}=K$ and $\mI_n^{\Inn (\mI_n)}=K$, the (unique) maximal ideal of $\mI_n$ is the {\em only} nonzero prime $\rG_n$-invariant ideal of $\mI_n$, and there are precisely $n+2$ $\rG_n$-invariant ideals of $\mI_n$. For each automorphism $\s \in \rG_n$, an {\em explicit inversion formula} is given via the elements $\s (\frac{\der}{\der x_i})$ and $\s (\int_i)$., Comment: 27 pages

We prove that the algebra $\mI_n:=K\langle x_1, ..., x_n, \frac{\der}{\der x_1},...,\frac{\der}{\der x_n}, \int_1, ..., \int_n\rangle $ of integro-differential operators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull dimension $n$ and of Gelfand-Kirillov dimension $2n$. Its weak homological dimension is $n$, and $n\leq \gldim (\mI_n)\leq 2n$. All the ideals of $\mI_n$ are found explicitly, there are only finitely many of them ($\leq 2^{2^n}$), they commute ($\ga \gb = \gb\ga$) and are idempotent ideals ($\ga^2= \ga$). The number of ideals of $\mI_n$ is equal to the {\em Dedekind number} $\gd_n$. An analogue of Hilbert's Syzygy Theorem is proved for $\mI_n$. The group of units of the algebra $\mI_n$ is described (it is a huge group). A canonical form is found for each integro-differential operators (by proving that the algebra $\mI_n$ is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra $\mA_n$ (but $\GK (\mA_n) =3n$, note that $\mI_n\subset \mA_n$). It is proved that the algebras $\mI_n$ and $\mA_n$ are ideal equivalent., Comment: 27 pages

The groups of automorphisms of the Jacobian algebras are found., Comment: 41 pages

Explicit generators are found for the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in $n$ variables. An analogue of the polynomial Jacobian homomorphism is found., Comment: 27 pages

Explicit generators are found for the group $G_2$ of automorphisms of the algebra $\mS_2$ of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that $$ G_2\simeq S_2\ltimes \mT^2\ltimes \Z\ltimes ((K^*\ltimes E_\infty (\mS_1))\boxtimes_{\GL_\infty (K)}(K^*\ltimes E_\infty (\mS_1)))$$ where $S_2$ is the symmetric group, $\mT^2$ is the 2-dimensional torus, $E_\infty (\mS_1)$ is the subgroup of $\GL_\infty (\mS_1)$ generated by the elementary matrices. In the proof, we use and prove several results on the index of operators, and the final argument in the proof is the fact that ${\rm K}_1 (\mS_1) \simeq K^*$ proved in the paper. The algebras $\mS_1$ and $\mS_2$ are noncommutative, non-Noetherian, and not domains. The group of units of the algebra $\mS_2$ is found (it is huge)., Comment: 13 pages