The algebra of one-sided inverses of a polynomial algebra

Abstract

a b s t r a c tWe study in detail the algebra S n in the title which is an algebra obtained from a polynomial algebra P n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P n . The algebra S 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 S n is 2n; but the weak and the global dimensions of S n are n. The prime and maximal spectra of S n are found, and the simple S n -modules are classified. It is proved that the algebra S n is central, prime, and catenary. The set I n of idempotent ideals of S n is found explicitly. The set I n is a finite distributive lattice and the number of elements in the set I n is equal to the Dedekind number d n .

An analogue of Hilbert’s Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra

“…Since F (Q) = Q ⊗ Z F (where F = F (Z)) is the least non-zero ideal of the algebra S 1 (Q) (Proposition 2.5. (2), [6]) and S 1 ⊆ S 1 (Q), it follows from (2)…”

Nonsplit module extensions over the one-sided inverse of k[x]

“…It is proved by Bavula [1] and Gerriten [2] that there is only one isomorphic class of infinite-dimensional simple R-modules. Note that there is an algebra monomorphism…”

K1() and the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in two variables

“…i / 2 K n g is the n-dimensional algebraic torus, Inn.S n / is the group of inner automorphisms of the algebra S n (which is huge), and GL 1 .K/ is the group of all the invertible infinite dimensional matrices of the type 1 C M 1 .K/ where the algebra (without 1) of infinite dimensional matrices The results of the papers [1,4,5,6] and the present paper show that (when ignoring the non-Noetherianness of S n ) the algebra S n belongs to a family of algebras including the n'th Weyl algebra A n , the polynomial algebra P 2n and the Jacobian algebra A n (see [1,6]). The structure of the group G 1 D T 1 Ë GL 1 .K/ is another confirmation of the 'similarity' of the algebras P 2 , A 1 , and S 1 ; they can be seen as 'almost' commutative polynomials.…”

“…We collect some results without proofs on the algebras S 1 and S 2 from [4] that will be used in this paper, their proofs can be found in [4].…”

“…Proposition 2.6 [4] The polynomial algebra P n is the only (up to isomorphism) faithful, simple S n -module.…”