Relations of the Algebra of Polynomial Integrodifferential Operators

Abstract

We revisit the algebra of polynomial integrodifferential operators and we give a generalization of its relations. Generators of prime ideals of height 1, of the maximal ideal, and of the smallest ideal of are discussed. A technique for obtaining a generating set for a given two-sided ideal of is also discussed.

1. Introduction

The algebra of polynomial integrodifferential operators was introduced in [1] and was studied extensively in [2–6]. Early in 2009, the author of [1] introduced the Jacobian algebra (see [7]) which has similar properties to the algebra of polynomial integrodifferential operators and both algebras are ideally equivalent. In 2010, the same author introduced the algebra (see [8]) of one-sided inverses of a polynomial algebra which is a generalized Weyl algebra as well as the Jacobian algebra and the algebra of polynomial integrodifferential operators . As applications, the algebra of polynomial integrodifferential operators has interesting applications in computer algebra as described in [9, 10]. Our goal in this work is to firstly study new relations of the algebra of polynomial integrodifferential operators in order to facilitate the use of this algebra in a computational point of view. Our starting point is to prove some relations introduced in [1] in Proposition 4. What is more, we give a generalization of these relations in Corollary 7 and then we give new other relations in Lemma 10. In the second part of this paper we study generators of the ideals of . Prime ideals of height 1 listed in [1] are revisited in Theorem 11, sets of generators of the only maximal ideal and of the smallest ideal of are studied, and a given two-sided ideal of is also discussed. The relations discussed in this paper can be used to revisit the special case of , and some results were obtained in the general case of Generalized Weyl algebras especially those described in [11–16].

2. Relations of

Let be a field and be a polynomial ring over . Denote .

Definition 1. By a Weyl algebra we mean the free associative algebra satisfying and where .

Definition 2. By the algebra of polynomial integro-differential operators we mean the algebra generated by the Weyl algebra and endomorphisms i.e

Remark 3. Observe that for Set ; regarding as the identity map of , we have the endomorphismDenoting , we see that ; that is, . From here, we recall Proposition 2.2 introduced in [1].

Proposition 4 (see Proposition 2.2, [1]). (1)The algebra is generated by the elements that satisfy the following defining relations: ,(a);(b);(c);(d);(e)For , we have .(2)The algebra is a Generalized Weyl algebra where and . Moreover, the algebra is graded where , where and such that(3)Each element of is a unique finite sum for unique elements .

Proposition 5 proves the first item of Proposition 4 left in [1].

Proposition 5. The algebra satisfies the following relations: (1);(2);(3);(4)(5);(6)For , we have .

Proof. (1).(2)By definition  Thus, .(3).(4).(5). Observe that  since .(6)Applying Remark 3, where and for all , then we have . Using now Definition 1, we see that

Setting , we easily see that where .

Example 6 (construction of ). Let us consider the polynomial ring in two indeterminates. By a monomial in we mean an element of the form where , and by a polynomial we mean a finite linear combination of monomials. Consider the following operators: O1O2O3O4O5O6The algebra of polynomial integrodifferential operators is the free associative algebra generated by the operators defined in O1,...,O6 with coefficients in and multiplication given by the composition of applications. The relations between operators are given by Proposition 5.

With notations as above we have the following result.

Corollary 7. For we have (1);(2);(3);(4);(5);(6);(7).

Proof. (1) For we have By Proposition 5 we have . Replacing this in the previous expression we haveObserve that Replacing this in the previous expression we have the following.By induction using the same technique we see that For we haveWe have seen that , and using this we haveBy induction we easily see that(2)Using Proposition 5 we see that . Applying the same techniques as in the previous item we see that (3) By Proposition 5 we see that and then Observe that and replacing this in we see thatBy we see that and combining and we have(4) Using the same technique as in the previous item and the fact that we see that (5)since we have shown above that . Observe that Replacing all this in we have the following result:Now let us prove that . Observe that Observe that , and using this in we have the following result: Now it becomes easy to see that (6) The same techniques as in the previous item can be easily applied.

Example 8 (application). Let be a univariate polynomial ring and let us consider the algebra constructed from ; we have . Setting , one can study the noncommutative polynomial ring . (1) Monomials in : in our case monomials are of the form where . Note that are not monomials but polynomials for since by Corollary 7 we have(i);(ii).(2) Polynomials in : a polynomial in is a linear combination of monomials. As a specific case, let us consider the polynomial . According to the order we have imposed to our variables, this polynomial is not well-organized. Observe that  after considering (1) in Proposition 5 and (1) in Corollary 7. Replacing in we get  Using (5) in Corollary 7 we get

We have seen in [1] that where and the only proper ideal of is .

Corollary 9. (1) The algebra is principal and its only proper ideal is ;
(2) is an module.

Proof. Consider the notation above.
(1) We have seen in [1] that the only proper ideal of is . Let us now prove that .
Let ; then Observe that each term of belongs in as a two-sided ideal; therefore .
The converse is straightforward since .
(2) can be regarded as an module. Moreover, We easily see that the external multiplication is only concern with .

Lemma 10. For and , (1);(2);(3);(4);(5).

Proof. (1)Seeing that algebra is associative and the fact that (by Proposition 4), then  Furthermore .(2). Observe that  But (3).(4).(5).(a)If then and , since then .(b)If , then and , since .

3. Two-Sided Ideals of

The theory of ideals of is already studied in [1]. The goal of this section is to study generating sets of those ideals and to study the behavior of a given ideal.

3.1. Generators of Primes Ideals of Height 1

Corollary 3.3 in [1] proved that the only prime ideals of of height 1 are

where .

With notations as above we have the following result.

Theorem 11. For we have for some .

Proof. By definition where If , then where and .
Observe that ; that is, where for . By Proposition 2.2 in [1], we see that where and .
It is easy to see as a two-sided ideal of . Observe that We then have
since ; i.e., if we haveThis means that for ; thus implies straightforwardly Using the same technique as above, we have Thus

Corollary 12. (1)The only maximal ideal of is  where .(2)The smallest nonzero ideal of is .

Proof. It is well known (see [1]) that (1)the only maximal ideal of is  that is, (2)the smallest nonzero ideal of is It is straightforward to see that . We prove the converse for since the general case can be obtained by induction. Let ; then . That is, for some . That is,for , and .
We have the following.Let us expand (58):since different variable commutes. Observe that
for ; this leads to . To avoid this, let us take ; in this case we have the following.Observe that
for some . This is because(i) and;(ii) since .Observe also that
for some (using Corollary 7).
Replacing this in we getSince different variable commutes, we have by Proposition 5(6), and this leads directly to . To avoid this case, we take ; in this case becomesAgain since different variable commutes, we have Replacing this in we getObserve the following.In any of these cases we finish with an expression of the form for some and ; thusLet us expand (59). Applying the same technique as in (58) we getwhere and .
In (68) we see that and in (69) ; thus

Example 13. Let us consider again the algebra . The maximal ideal of is where and for some polynomials and . Thus . The smallest ideal is .

3.2. Generators of Ideals of

Let be a two-sided ideal of ; then can be written as .

For , let ; then

It is proven in [1], Proposition 2.2, that where and For we have where and . Thus we have the following.

(1) If each , then

Observe that

and ; this implies the following.

If

note that

and then

(2) Assume without loss of generalities that there exist such that

and in this case

Remark 14. Obtaining a set of generators of a given two-sided ideal of is very important. According to the theory of ideals studied in [1], we see that any two-sided ideal of can be written as a product or intersection of primes ideals of . A very interesting result will be to propose an algorithm for finding these prime ideals from a given ideal. This work proposes the first step of tools necessary to deal with this problem, but to propose a decomposition algorithm, we will need to study the theory of two-sided Groebner bases (see [17]) on as well as some applications such as the two-sided module of syzygies, since it is very important for the computation of generating sets of intersection of ideals as described in [18]. Our next project will develop computer algebra in and solve this problem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.