An algebraic study of multivariable integration and linear substitution

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

The Jacobson radical of inverse Laurent power series of rings

For a ring [Formula: see text] with a derivation [Formula: see text], we determine the Jacobson radical of the pseudo-differential operator ring [Formula: see text] for a large class of rings with a [Formula: see text]-condition. Various types of examples of these rings are provided.

On Annihilator Ideals of Pseudo-differential Operator Rings

Let R be a ring with a derivation δ and R((x-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

On prime one-sided ideals, bi-ideals and quasi-ideals of a gamma ring

Let M be a Γ-ring with right operator ring R. We define one-sided ideals of M and show that there is a one-to-one correspondence between the prime left ideals of M and R and hence that the prime radical of M is the intersection of its prime left ideals. It is shown that if M has left and right unities, then M is left Noetherian if and only if every prime left ideal of M is finitely generated, thus extending a result of Michler for rings to Γ-rings.Bi-ideals and quasi-ideals of M are defined, and their relationships with corresponding structures in R are established. Analogies of various results for rings are obtained for Γ-rings. In particular we show that M is regular if and only if every bi-ideal of M is semi-prime.