Uniform Dimension over Skew PBW Extensions

The aim of the present paper is to show that, under some conditions, the uniform dimension of a ring R is the same as the uniform dimension of a skew Poincaré-Birkhoff-Witt extension built on R.


Introduction
A basic tool in the study of Noetherian rings and modules is the uniform dimension (also known as Goldie dimension), noted rudim(−) for the right dimension (similarly ludim(−) for the left dimension).The basic idea of this dimension is that one measures the "size" of a module M by finding out how big a direct sum of nonzero submodules M can contain.For modules over a division ring, uniform dimension is just the usual vector space dimension as defined in linear algebra.
For polynomial rings, Shock in 1972 ([15], Theorem 2.6) proved that if B is a ring having finite left uniform dimension, then the left uniform dimension a Seminario de Álgebra Constructiva -SAC2.
of B[x] is equal to the left uniform dimension of B (see also Goodearl [3], Theorem 3.23).In the case of noncommutative rings, and more specifically skew polynomial rings, we can include (in chronological order) the following works: In 1988, Grzeszczuk [5] proved that if B is a semiprime left Goldie ring equipped with a derivation δ, then the Goldie dimension of B[y; δ] is equal to the Goldie dimension of B. In fact, he proved that B[y; δ] B[y;δ] and B B have the same uniform dimension if B is right nonsingular, or if B is a Q-algebra with the descending chain condition on right annihilators ( [5], Corollary 4).The same year, Quinn ([12], Theorem 15) showed that if B is a Q-algebra and δ is locally nilpotent, then B[y; δ] B[y;δ] and B B have the same uniform dimension.This result cannot hold in general; the classical example is given by B = k[x]/ x 2 and δ = d dx , where k is a field of characteristic 2, in which case rudim(B B ) = 1 and rudim(B[y; δ] B[y;δ] ) = 2 ([4], p. 851).In 1995, Matczuk [9] proved that if B is a semiprime left Goldie ring equipped with an automorphism σ and σ-derivation δ, then the Goldie dimension of B[x; σ, δ] is equal to the Goldie dimension of B. In 2005, Leroy and Matczuk [7] generalized this result to the case where σ is an injective endomorphism.A similar remark can be established for the results presented by Mushrub [11] and Sigurdsson [16].
In this paper we present sufficient conditions to guarantee that a ring R and a skew Poincaré Birkhoff Witt extension A built on R have the same uniform dimension.Since skew P BW extensions introduced in [2] are a generalization of P BW extensions, the results established here are more general than the result presented in [1].In this way this paper continues with the study of several dimensions of skew P BW extensions presented in [8], Section 4, [13] and [14], Chapter 4. The techniques used here are fairly standard and follow the same path as other text on the subject.The results presented are new for skew P BW extensions and all they are similar to others existing in the literature.
The paper is organized as follows.Section 2 contains the definition and some of the properties of the objects we are going to study.In Section 3 we establish an upper bound for the uniform dimension of skew P BW extensions, and in Section 4 we present sufficient conditions under which passing from R to A preserves the dimension.For example, if M is a nonsingular right R-module, or if each nonzero submodule of M contains a nonzero element whose annihilator in R is (Σ, ∆)-invariant, then M ⊗ R A has the same uniform dimension as M .When R is right Noetherian ring and tame as a right module over itself and with prime annihilator ideals under certain conditions of stability, we show that the uniform dimension of both A A and R R coincides.
Throughout this paper the rings and algebras are associative with unit, and all modules are unital right modules.

Definitions and Elementary Properties
In this section we recall the definition of skew P BW extensions presented in [2] and we also present some key properties of these extensions.The content and proofs of this introductory section can be found in [8], Sections 1 and 2, or [14], Chapter 1. From Definition 2.1 we can see that skew P BW extensions are a generalization of P BW extensions defined by Bell and Goodearl in [1] (see [2] for more details).Definition 2.1 ([2] Definition 1).Let R and A be rings.We say that A is a skew P BW extension of R (also called a σ-P BW extension of R) if the following conditions hold: (ii) There exist elements x 1 , . . ., x n ∈ A R such that A is a left free Rmodule, with basis the basic elements (iii) For each 1 ≤ i ≤ n and any r ∈ R {0}, there exists an element c i,r ∈ R {0} such that (iv) For any elements 1 ≤ i, j ≤ n there exists c i,j ∈ R {0} such that Under these conditions we will write A := σ(R) x 1 , . . ., x n .
(i) Since Mon(A) is a left R-basis of A, the elements c i,r and c i,j in Definition 2.1 are unique.
(ii) In Definition 2.1 (iv), c i,i = 1.This follows from and hence 1 = c j,i c i,j , that is, for each 1 ≤ i < j ≤ n, c i,j has a left inverse and c j,i has a right inverse.In general, the elements c i,j are not two sided invertible.For instance, we cannot conclude that c 12 c 21 = 1 because x 2 x 1 is not a basic element of Mon(A) (we recall that Mon(A) consists of the standard monomials).
(iv) Each element f ∈ A {0} has a unique representation as The next proposition justifies the notation and the name of the skew P BW extensions.
For each 1 ≤ i ≤ n, there exist an injective endomorphism σ i : R → R and a σ i -derivation δ i : R → R such that A particular case of skew P BW extension is considered when derivations δ i are zero for all i.A remarkable case is presented when all endomorphisms σ i are isomorphisms.These observations are formulated in the next definition.(a) A is called quasi-commutative if the conditions (iii) and (iv) in Definition 2.1 are replaced by (iii') for each 1 ≤ i ≤ n and all r ∈ R {0} there exists c i,r ∈ R {0} such that (iv') for any 1 ≤ i, j ≤ n there exists c i,j ∈ R {0} such that (ii) For X = x α ∈ Mon(A), exp(X) := α and deg(X) := |α|.The symbol will denote a total order defined on Mon(A) (a total order on N n 0 ).For an element ) is the order of f , and Finally, if f = 0, then lm(0) := 0, lc(0) := 0, lt(0) := 0. We also consider X 0 for any X ∈ Mon(A).For a detailed description of monomial orders in skew P BW extensions, see [2,Section 3].
Skew P BW extensions can be characterized as the following theorem shows.
Theorem 2.7 ([2], Theorem 7).Let A be a polynomial ring over R with respect to {x 1 , . . ., x n }.A is a skew P BW extension of R if and only if the following conditions are satisfied: (i) for each x α ∈ Mon(A) and every nonzero element r of R, there exist unique elements where where c α,β is left invertible, p α,β = 0 or deg(p α,β ) < |α + β| if p α,β = 0.
In the noncommutative setting an integral domain, briefly called a domain, is defined as a ring in which the product of any two nonzero elements is nonzero.With this in mind, if A is a skew P BW extension of a domain R, then so is A ([8, Proposition 4.1]).
Skew P BW extensions are filtered rings.We recall the definition of these rings.
Definition 2.8.A filtered ring is a ring B with a family F B = {F n B : n ∈ Z} of additive subgroups of B where we have the ascending chain From a filtered ring B it is possible to construct its associated graded ring G(B) which is known in the literature as the associated graded ring of B.
The first key theorem computes the graduation of a general skew P BW extension of a ring R. Theorem 2.9 ([8], Theorem 2.2).Let A be an arbitrary skew P BW extension of R.Then, A is a filtered ring with filtration given by and the corresponding graded ring Next we recall the Hilbert's Basis theorem for skew P BW extensions.
Theorem 2.10 ([8], Corollary 2.4).Let A be a bijective skew The next theorem is also very useful in the following section.
Proposition 2.11.If A is a bijective skew P BW extension of a prime ring R, then A is also a prime ring.
Proof.Theorem 2.9 shows that G(A) is a quasi-commutative skew P BW extension of R, and by assumption The result follows from [10, Theorem 1.2.9 and Proposition 1.6.6].

Uniform Dimension over Skew P BW Extensions I
In this section we establish a relation between the uniform dimensions of a ring R and a skew P BW extension A built on R. If A is a bijective skew P BW extension of a right Noetherian domain R we will show that rudim A = rudim R = 1.In a more general case, we prove that if A is a bijective skew P BW extension of a prime right Goldie ring R, then the uniform dimension of A is bounded by the uniform dimension of R.
Definition 3.1.Let B be a ring.If N is a submodule of a right B-module M such that, for all nonzero submodules X of M , one has N ∩ X = 0, then N is an essential submodule of M , and M is an essential extension of N .We write A module U is uniform if U = 0 and each nonzero submodule of U is an essential submodule.This is equivalent to U not containing a direct sum of nonzero submodules.For example, if B is an integral domain, then B B is uniform if and only if B is a right Ore domain.We recall that a module M is said to have finite uniform dimension if it contains no infinite direct sum of nonzero submodules.This is true of any uniform module and of any Noetherian module.Note that a module with Krull dimension has finite uniform dimension ([10, Lemma 6.In order to determine an upper bound for a bijective skew P BW extension we need the following lemma.We thank professor Huishi Li for a personal communication with a simplification of our original proof.Before, we recall that if B is a filtered ring with filtration F B = {F n B} n∈Z and M is a right B-module, the induced filtration For elements r ∈ G(R) n and y ∈ G(N i ) m we define (r +F n−1 R)(y +F m−1 N i ) = ry+F n+m−1 and thus G(N i ) is a graded submodule of G(M ) which gives rise to a direct sum of graded submodules of G(M ).If the filtration F M is separated and exhaustive, then G(N i ) = 0 if and only if N i = 0.The result follows from [10, Theorem 2.2.9].Theorem 3.5.Let R be a prime right Goldie ring.If A is a bijective skew P BW extension of R, then uniform dimension of A is less or equal than uniform dimension of R.
Proof.By Lemma 3.4 we obtain rudim A ≤ rudim G(A).Theorem 2.11 and Proposition 3.3 (this last says that the uniform dimension is preserved by iterated polynomial rings of automorphism type), imply that rudim G(A) = rudim R.

Uniform Dimension over Skew Quantum Polynomials
In this section we compute the uniform dimension of skew quantum polynomials introduced in [8].
Let R be a ring with a fixed matrix of parameters q := [q ij ] ∈ M n (R), n ≥ 2, such that q ii = 1 = q ij q ji = q ji q ij for every 1 ≤ i, j ≤ n, and suppose that automorphisms σ 1 , . . ., σ n of R are also given.The ring of skew quantum polynomials over R, denoted by R q,σ x ±1 1 , . . ., x ±1 r , x r+1 , . . ., x n or Q r,n q,σ (R) is defined as the ring satisfying the relations: 1 , . . ., x ±1 r , x r+1 , . . ., x n ; (ii) R q,σ x ±1 1 , . . ., x ±1 r , x r+1 , . . ., x n is a free left R-module with basis (iii) the variables x 1 , . . ., x n satisfy the defining relations Remark 3.7.R q,σ x ±1 1 , . . ., x ±1 r , x r+1 , . . ., x n can be viewed as a localization of a skew P BW extension.For the quasi-commutative bijective skew P BW extension A := σ(R) x 1 , . . ., x n , with x i r = σ i (r)x i and x j x i = q ij x i x j , 1 ≤ i, j ≤ n.If we set S := {rx α : r ∈ R * , x α ∈ Mon{x 1 , . . ., x r }}, then S is a multiplicative subset of A and we have the isomorphism  Proof.The assertion follows from Remark 3.7, Proposition 3.2 and Lemma 3.9.
Proof.The result follows from Remark 3.7, Theorem 3.5 and Lemma 3.9.

Uniform Dimension over Skew P BW Extensions II
In this section we establish sufficient conditions under which passing from R to A preserves the uniform dimension for A a bijective skew P BW extension of R.
Let A = σ(R) x 1 , . . ., x n be a skew P BW extension of a ring R. By Proposition 2.3 we know that x i r − σ i (r)x i = δ i (r) for all r ∈ R, where σ is an injective endomorphism of R and δ i is a σ i -derivation of R for each 1 ≤ i ≤ n.Let Σ := {σ 1 , . . ., σ n } and ∆ := {δ 1 , . . ., δ n }.We say that the pair (Σ, ∆) is induced by the variables x 1 , . . ., x n .If I is an ideal of R, I is called Σ-invariant (∆-invariant) if it is invariant under each injective endomorphism (σ-derivation) of Σ (∆), that is, σ i (I) ⊆ I (δ i (I) ⊆ I) for 1 ≤ i ≤ n.If I is both Σ and ∆invariant ideal we say that I is (Σ, ∆)-invariant.We consider a (Σ, ∆)-invariant ideal I of R to be (Σ, ∆)-prime if whenever a product of two (Σ, ∆)-invariant ideals is contained in I, one of these ideals is contained in I. R is a (Σ, ∆)-prime ring if the ideal 0 is (Σ, ∆)-prime.
The next proposition is very useful for computing uniform dimension of skew P BW extensions.(ii) It is also clear that A/IA is a left R/I-module with generating set Mon(A/IA).Next we show that A/IA is a left free R/I-module.Consider the expression for each i.Let us see that r i = 0 + I for each i.By definition above we have A is a left free R-module, by order conditions on X i using notation in Definition 2.6 we can write Thus we obtain that r i = m i for all i which implies that r i ∈ I and thus r i = 0 + I for i = 1, . . ., n. Therefore A/IA is a left free R/I-module.
(iii) Let r = 0 + I.We have x i r = x i r = 0 + AI since r / ∈ I. Then x i r / ∈ IA for each i.By Proposition 2.3 we know that x i r = c i,r x i + δ i (r) for all r ∈ R and each i.Since R is left Noetherian, for every σ ∈ Σ we obtain I = σ(I).Then, if r / ∈ I it follows that c i,r = σ i (r) / ∈ I.In this way c i,r x i / ∈ IA whence δ i (r) / ∈ IA which yields δ i (r) / ∈ I for 1 ≤ i ≤ n.Therefore we consider x i r = c i,r x i + δ i (r), i = 1, . . ., n.Since Mon(A/IA) is a R/I basis of A/IA then c i,r is unique (Remark 2.2).
(iv) Note that x j x i = 0 + IA since x j x i / ∈ IA for 1 ≤ i < j ≤ n.By assumption, the elements c i,j are left invertible in R which implies that c i,j / ∈ I and thus c i,j x i x j / ∈ IA for 1 ≤ i < j ≤ n.Hence x j x i − c i,j x i x j = n t=1 r t x t / ∈ IA, where r t ∈ R. Since A is a left free R-module, there exists j ∈ {1, . . ., n} with r j / ∈ I and thus r j x j / ∈ IA.Thus n t =j r t x t / ∈ IA.Continuing this way we can see that r t / ∈ I for all t = 1, . . ., n, and we obtain the equality x j x i = c ij x i x j + n t=1 r t x t , where c i,j = 0 + I, x i x j = 0 + IA and r t = 0 + I for all 1 ≤ i < j ≤ n and t = 1, . . ., n, respectively.Since Mon(A/IA) is a R/I basis of A/IA the elements c i,j are unique (see Remark 2.2).
In this way A/IA is a skew P BW extension of R/I.We keep the variables x 1 , . . ., x n of extension A of the extension A/IA hoping that this will not cause confusion.
If M is a right R-module, and T is a nonzero A-submodule of M ⊗ R A, since R A is free, whence faithfully flat, given any right R-modules N ≤ M , we may identify N ⊗ R A with its image in M ⊗ R A. The module M ⊗ R A is called the induced module.Observe that M ⊗ R A is, as an abelian group, the direct sum of the subgroups M ⊗ X i for each X i ∈ Mon(A).In this way, any nonzero element f ∈ M ⊗ R A may be uniquely expressed in the form where m i ∈ M for each i, m t = 0, and exp(X i ) ≺ exp(X t ), 1 ≤ i ≤ t − 1.We shall usually abbreviate such an expression to  Proof.Let x, y ∈ M ⊗ R A with x = 0. Consider the elements and where x i , y j ∈ M , x t , y s = 0, exp(x) := exp(X t ), and exp(y) := exp(X s ).For k = s, s − 1, . . ., 0, the idea is to show that there exists r k ∈ R such that x t r k = 0 and With this in mind, since N ≤ r M there exists r s ∈ R such that x t r s = 0 and y s r s ∈ N .Because A is bijective, let r s := σ − exp(X s ) (r s ).Following notation (14), Theorem 2.7 (i) yields or, equivalently, with p exp(X l ),r s ∈ A for all l = 1, . . ., t, and p exp(X l ),r s = 0, or deg , with exp p exp(X l ),r s := exp X l,h(l) , and the d l 's are elements of R, the X l 's are basic elements of Mon(A), and the value h(l) depends of the polynomial l.Then In this way, from (17) This shows that for the element yr s we have the sets of basic monomials given by X 1 , X 2 , . . ., X s , X 1,1 , X 1,2 , . . ., X 1,h(1) , X 2,1 , X 2,2 , . . ., X 2,h(2) , . .., X s,1 , X s,2 , . . ., X s,h(s) .Of course, these sets are not necessarily disjoint (note that exp(X s ) is greater than others basic elements of yr s ).If we consider the union after suppressing possible repetitions of basic monomials, we have a finite number of monomials X 1 , . . ., X v−1 , X s , say, if no confusion arises with (16).So, from the last expression for yr s above, we obtain Let 0 < k ≤ s.Suppose that there exists r k ∈ R which satisfies the required properties.Consider the expression with z 0 , . . ., z k−1 ∈ M and z k , . . ., z s ∈ N .There exists p ∈ R such that x t r k p = 0 and z k−1 p ∈ N .Therefore the element r k−1 = r k p has the required properties.In this way we complete the inductive step.Then x t r 0 = 0 which implies xr 0 = 0 and yr Remark 4.4.In the proof of Lemma 4.3 we assume that the skew P BW extension is bijective.Nevertheless, we only used the fact that the injective endomorphisms σ of Proposition 2.3 are bijective, that is, we do not require that the elements c i,j are invertible.
For the next lemma consider a bijective skew P BW extension A of a ring R, M a right R-module, and T a nonzero A-submodule of M ⊗ R A.
Lemma 4.5.If f is a nonzero element of T of minimal monomial order exp(X t ) = α t among all elements of T (f is expressed as in (13)), then σ −αt rann R lc(f ) A = rann A (f ).Thus f A ∼ = lc(f )R ⊗ R A as right Amodules.

Let us see now that rann
Repeating this process we show that the summands . ., r 0 are elements of σ −αt rann R lc(f ) A which yields that u ∈ σ −αt rann R lc(f ) A and hence we prove the inclusion rann Proof.If R is a right Noetherian ring or M is a Noetherian module, then every nonzero submodule of M contains a uniform Noetherian submodule.This implies that M contains an essential submodule N which is a direct sum of uniform Noetherian submodules.Since M is nonsingular, N ≤ r M and so by Lemma In this way we have to show that if M is a nonsingular uniform Noetherian module, then M ⊗ R A is uniform.Since M ⊗ R A is Noetherian, it contains a uniform submodule T .Consider an element nonzero f of T of minimal monomial order as in Lemma 4.5, Lemmas 4.3 and 4.5 imply that The next proposition establishes that nonsingularity is preserved for induced modules.
Proposition 4.8.Let A be a bijective skew P BW extension of a ring R and let M be a right R-module.
Proof.Suppose that M R is nonsingular.Let T be the singular submodule of M ⊗ R A. If T = 0, let f ∈ T be nonzero with minimal monomial order as in Lemma 4.5.We obtain that rann A (f ) = rann R lc(f ) A,and since M is nonsingular, there is a nonzero right ideal I of R with rann R lc(f ) ∩ I = 0. Hence rann R lc(f ) A ∩ IA = 0 which implies that rann A (f ) is not an essential right ideal of A, which contradicts the definition of T .We conclude that T = 0.
Finally suppose that R R and (M ⊗ R A) A are nonsingular.Let m be an element of M with I = rann R (m).If I is an essential right ideal of R, then I R ≤ r R R and hence IA A ≤ r A A .The fact (m ⊗ 1)IA = 0 implies that m = 0 which shows that M R is nonsingular.Definition 4.9 ([1], Section 2).Let B be a right Noetherian ring and let U be a uniform right B-module.Then there is a unique prime ideal P of B which is the largest annihilator of any nonzero submodule of U .This prime ideal is called the assassinator of U , and U is called tame if it contains a copy of a nonzero right ideal of B/P .Alternatively, U is tame if and only if the submodule rann U (P ) is torsion free as an (B/P )-module.An arbitrary right B-module M is tame if all of its uniform submodules are tame, and we denote the set of assassinator prime ideals of uniform submodules of M by ass(M ).Proposition 4.10.Let A be a bijective skew P BW extension of a right Noetherian ring, let (Σ, ∆) be the pair induced by x 1 , . . ., x n and let M be a tame right R-module such that each member of ass(M ) is (Σ, ∆)-invariant.Then rudim R (M ) = rudim A (M ⊗ R A).
Proof.Let E be the injective hull of M .Since it is sufficient to show that rudim R (E) = rudim A (E ⊗ R A).Since R is right Noetherian, E is a direct sum of uniform (indecomposable) injective submodules.Using the fact that the tensor product preserves direct sums, it is enough to prove the assertion with E uniform ([6, Theorem 3.48 and Corollary 6.10]).We also note that neither the tameness of M nor the set ass(M ) is changed by passing to an essential extension or an essential submodule of M ([1, p. 20]).In this way, following Definition 4.9 we may consider the case where M = E(U ) is the injective hull of a uniform right ideal U of some factor ring R/P with P a (Σ, ∆)-invariant prime ideal of R.
Let E 0 = ann E (P ).Then E 0 is the (R/P )-injective hull of U , and E 0 is torsionfree and uniform as an (R/P )-module, so by Proposition 4.7 the module E 0 ⊗ R/P (A/P A) ∼ = E 0 ⊗ R A is uniform as a right A-module (note that A/P A is a skew P BW extension of R/P by Proposition 4.1).In this way, to conclude the proof we have to show that E 0 ⊗ R A ≤ e E ⊗ R A. By contradiction, suppose that E 0 ⊗ R A is not essential in E ⊗ R A. Then there is a nonzero element a ∈ E ⊗ R A of minimal monomial order such that aA ∩ (E 0 ⊗ R A) = 0. Following (13) we have the expression a = (a 0 ⊗ 1) + (a 1 ⊗ X 1 ) + where a i ∈ E for each i, a m = 0, exp(X i ) ≺ exp(X m ), 1 ≤ i ≤ m − 1, and the element a satisfies the conditions of the Lemma 4.5.Since E 0 is essential in E, there exists r ∈ R such that a m r ∈ E 0 and a m is nonzero.We may replace a by ar and then without lost of generality we suppose that a m ∈ E 0 .In this way a m P = 0, and using the fact that P is (Σ, ∆)-invariant and part (i) of Theorem 2.7 we have that (a m ⊗X m )P = 0. Now, the equality aA∩(E 0 ⊗ R A) = 0 implies aP A ∩ (E 0 ⊗ R A) = 0, and using the minimality of m we obtain that aP = 0 whence a − (a m ⊗ X m ) P = 0. Thus a m−1 P = 0. Continuing this way we can see that a i P = 0 for every a i , but this means that a ∈ E 0 ⊗ R A, which contradicts a = 0. So, E 0 ⊗ R A ≤ e E ⊗ R A and the assertion follows.
Next theorem establishes conditions under which passing from R to A preserves the dimension where A is a skew P BW extension of R. Theorem 4.11.Let A be a bijective skew P BW extension of a right Noetherian ring.Suppose that R is tame as a right R-module over itself and that any prime annihilator ideal in R is (Σ, ∆)-invariant.Then rudim R (R) = rudim A (A).
Proof.The assertion follows from Definition 4.9 and Proposition 4.10.
2.6]).Because bijective skew P BW extensions have Krull dimension ([8, Section 4]) these extensions have uniform dimension.Since a right Noetherian domain has right uniform dimension 1, Theorem 2.10 and [8, Proposition 4.1], yield the following proposition.Proposition 3.2.If A is a bijective skew P BW extension of a right Noetherian domain R, then the uniform dimension of A is 1, that is, rudim A = 1.A more general result than Proposition 3.2 is established in Theorem 3.5.Proposition 3.3 ([7], Theorem 3.4).If B is a semiprime right Goldie ring and σ is injective, then the Ore extension B[x; σ, δ] is also semiprime right Goldie and both rings have the same right uniform dimension.
where X is any system of generators of M .Lemma 3.4.Let B be a filtered ring and M a right B-module.Suppose that the induced filtration F M on M is separated and exhaustive.If rudim Gr(M ) = s, then rudim(M ) ≤ s.In particular, if B is filtered with separated and exhaustive filtration, then rudim B ≤ rudim G(B).Proof.Let B be a filtered ring with filtration F B = {F n B} n∈Z .Consider G(B) = n∈Z G(B) n , the associated graded ring of B, where we know that

Examples 3 . 8 .
Particular examples of skew polynomial rings include quantum polynomials, algebra of skew quantum polynomials, algebra of quantum polynomials, the n-multiparametric skew quantum space, n-multiparametric skew quantum torus, skew Laurent polynomial ring, n-multiparametric skew quantum torus, etc.For a detailed description of these rings and algebras, see[8, Example 3.2]  or[13, Remark 22].Lemma 3.9 ([10], Lemma 2.2.12).Let S be a left Ore set of regular elements of a ring B. Then rudim S B = rudim B. Proposition 3.10.If R is a right Noetherian domain, then ludim Q r,n q,σ (R) = 1.

Proposition 4 . 1 .
Let R be a right Noetherian ring and let A = σ(R) x 1 , . . ., x n be a bijective skew P BW extension of R. If I is a nonzero (Σ, ∆)-invariant ideal of R then IA = AI is an ideal of A with IA ∩ R = I, R/I embeds in A/IA and A/IA is a skew P BW extension of R/I.Proof.Since I is a (Σ, ∆)-invariant ideal of R it follows that IA = AI is an ideal of A with IA ∩ R = I.Let us see that A/IA is a skew P BW extension of R/I.(i)It is clear that R/I ⊆ A/IA.

)
Definition 4.2.A B-module M is a rational extension of a submodule N , denoted N ≤ r M , provided that Hom B (L/N, M ) = 0 for any submodule L of M that contains N .Equivalently, if these are right modules, N ≤ r M if and only if whenever x, y ∈ M with x = 0, there exists r ∈ R such that xr = 0 and yr ∈ N ([3, Proposition 2.25]).

Lemma 4 . 3 .
Let A be a bijective skew P BW extension of a ring R.If N ≤ r M are right R-modules, then N ⊗ R A ≤ r M ⊗ R A as R-modules and hence also as A-modules.

Definition 4 . 6 .
If M is a right module over a ring B, an element of m ∈ M is said to be a singular element of M if the right ideal rann B (m) is essential in B B .The set of all singular elements of M is denoted by Z(M ).M B is a singular (nonsingular ) module if Z(M ) = M (Z(0) := 0).We have the following key result.Proposition 4.7.Let A be a bijective skew P BW extension of a ring R and let M be a nonsingular right R-module.If either R is a right Noetherian ring or M is a Noetherian module, then rudim R (M ) = rudim A (M ⊗ R A).