Throughout this article, all rings considered are commutative with unity, all modules are unital and S always is a multiplicative subset of R, that is, 1∈ S and s1s2∈S for any s1∈S, s2∈S. Let R be a ring and M an R-module. Recall from Zhang, [3],that an R-module M is said to be uniformly S-torsion if sT=0 for some s ∈ S. The abbreviateion u- will always stand for 'uniformly'. An R-module M is S-finite if and only if M/F is u-S-torsion for some finitely generated submodule F of M. In the same way, Zhang defined an R-sequence 0→M→fN→gL→0 to be u-S-exact (at N) provided that there is an element s ∈ S such that sKer(g)⊆ Ima(f) and sIma(f)⊆Ker(g). We say a long R-sequence …→An−1→fnAn→fn+1An+1→… is u-S-exact, if for any n there is an element s ∈ S such that sKer(fn+1)⊆Ima(fn) and sIma(fn)⊆Ker(fn+1). A u-S-exact sequence 0→A→B→C→0 is called a short u-S-exact sequence. An R-homomorphism f:M→N is a u-S-monomorphism (resp., u-S-epimorphism, u-S-isomorphism) provided 0→M→fN (resp., M→fN→0, 0→M→fN→0) is u-S-exact. It is easy to verify an R-homomorphism f:M→N is a u-S-monomorphism (resp., u-S-epimorphism, u-S-isomorphism) if and only if Ker(f) (resp., Coker(f), both Ker(f) and Coker(f)) is a u-S-torsion module.

In [3], Zhang introduced the class of u-S-flat modules F for which the functor F⊗R− preserves u-S-exact sequences. The class of u-S-flat modules can be seen as a “uniform” generalization of that of flat modules, since an R-module F is u-S-flat if and only if Tor1R(F,M) is u-S-torsion for any R-module M. The class of u-S-flat modules has the following u-S-hereditary property: let 0→A→B→C→0 be a u-S-exact sequence, if B and C are u-S-flat so is A (see [[3], Proposition 3.4]).

In [5], the author introduced the u-S-flat dimensions of modules and rings. Let R be a ring, S a multiplicative subset of R and n be a positive integer. We say that an R-module has a u-S-flat dimension less than or equal to n, sfdR(M)≤n, if Torn+1R(M,N) is u-S-torsion R-module for all R-modules N. Hence, the u-S-weak global dimension of R is defined to be

As in [4], a u-S-exact sequence of R-modules 0→A→B→C→0 is said to be u-S-pure provided that for any R-module M, the induced sequence 0→M⊗RA→M⊗RB→M⊗RC→0 is also u-S-exact, and a submodule A of B is called a u-S-pure submodule if the exact sequence 0→A→B→B/A→0 is u-S-pure exact.

In [3], Zhang defined the u-S-von Neumann regular ring as follows: Let R be a ring and S a multiplicative subset of R. R is called a u-S-von Neumann regular ring provided there exists an element s ∈ S such that for any a ∈ R there exists r∈ R with sa=ra². Thus by [[3], Theorem 3.13], R is a u-S-von Neumann regular ring if and only if every R-module is u-S-flat.

In Section 2, we introduce the concept of w-u-S-flat modules and we study some characterization of w-u-S-flat modules. Hence, we prove that a ring R is u-S-von Neumann regular if and only if every R-module is w-u-S-flat. We prove also, if an R-module F is w-u-S-flat, then F_S is flat over R_S. A new local characterization of flat modules also is given. Section 3 deals with the w-u-S-flat dimension of modules and rings. After a routine study of these dimensions, we prove that R is a u-S-von Neumann regular ring if and only if w-u-S-w.gl.dim(R)=0 if and only if every R/I is w-u-S-flat for any ideal I of R.

In this section, we introduce a class of modules called weak u-S-flat modules and we study their properties and give their characterizations. The abbreviation w- always stands for 'weak'. We start with the following definition.

Definition 2.1. An R-module M is said to be w-u-S-flat if Tor1R(R/I,M) is u-S-torsion for any ideal I of R.

Obviously, every u-S-flat module is w-u-S-flat. If S is consist of units, then w-u-S-flat modules and u-S-flat modules coincide.

Remark 2.2. Let R=ℤ the ring of integers, p a prime in ℤ and S={pn|n≥0}. Let M=ℤ(p)/ℤ be a ℤ-module where ℤ(p) is the localization of ℤ at S. By Exapmle [[3], Example 3.3], we have M is w-u-S-flat but not u-S-flat.

Recall from [[2], Theorem 2.5.6], that an R-module M is flat if and only if for any (finitely generated) ideal I of R, 0→I⊗RM→R⊗RM is exact if and only if for any (finitely generated) ideal I of R, the natural homomorphism 0→I⊗RM→IM is an isomorphism. We give a u-S-analogue of this result.

Proposition 2.3. Let R be a ring, S be a multiplicative subset of R, and M be an R-module. The following are equivalent:

• 1. M is w-u-S-flat.

• 2. Tor1R(R/I,M) is u-S-torsion for any finitely generated ideal I of R.

• 3. The natural homomorphism I⊗RM→R⊗RM is a u-S-monomorphism, for any ideal I of R,.

• 4. The natural homomorphism I⊗RM→R⊗RM is a u-S-monomorphism, for any finitely generated ideal I of R.

• 5. The natural homomorphism μI:I⊗RM→IM is a u-S-isomorphism, for any ideal I of R.

• 6. The natural homomorphism μI:I⊗RM→IM is a u-S-isomorphism, for any finitely generated ideal I of R.

Proof. The implications (1)⇒(2), (3)⇒(4) and (5)⇒(6) are obvious.

(1)⇔(3) and (2)⇔(4). Let I be a (finitely generated) ideal of R. Then we have a long exact sequence:

Consequently, Tor1R(R/I,M) is u-S-torsion if and only if I⊗RM→R⊗RM is a u-S-monomorphism.

(3)⇒(5) and (4)⇒(6). Let I be a (finitely generated) ideal of R. Then we have the following commutative diagram:

Then, μI is a u-S-monomorphism. Since the multiplicative map μI is an epimorphism, μI is a u-S-isomorphism.

(6)⇒(3). Let I be an ideal of R. We just need to show Ker(μI) is u-S-torsion. Suppose that Ker(μI), ai∈I, xi∈M. Let I0=Ra1+…+Ran. Hence, I0⊆I. Consider the following commutative diagram:

By (6), μI0 is u-S-isomorphism. Thus, there exists s∈ S such that sΣi=1nai⊗xi=0 in I0⊗RM. Since h is a monomorphism, g is a u-S-monomorphism. Hence, there exists s'∈ S such that s′Σi=1nai⊗xi=0 in I⊗RM, which implies that Ker(μI) is u-S-torsion.

Corollary 2.4. Let R be a ring, S be a multiplicative subset of R and M be an R-module. The class of w-u-S-flat R-modules is closed under u-S-isomorphisms.

Proof. Let f:M→N be a u-S-isomorphisms, and I be an ideal of R. There exists two exact sequence 0→T1→M→L→0 and 0→L→N→T2→0 with T₁ and T₂ u-S-torsion. Consider the induced two long exact sequence, Tor1R(R/I,T1)→ Tor1R(R/I,M)→ Tor1R(R/I,L)→R/I⊗RT1 and Tor2R(R/I,T2)→ Tor1R(R/I,L)→ Tor1R(R/I,N)→ Tor1R(R/I,T2). By [[3], Corollary 2.6], M is w-u-S-flat if and only if N is w-u-S-flat.

Proposition 2.5. Let R be a ring, S be a multiplicative subset of R. R is u-S-von Neumann regular ring if and only if every R-module of R is w-u-S-flat.

Proof. ⇒. By [[3], Theorem 3.13].

⇐. Let I and J be ideals of R. We have Tor1R(R/I,R/J) u-S-torsion since R/J is w-u-S-flat. Thus, there exsits s∈ S such that sTor1R(R/I,R/J)=0. So, R is u-S-von Neumann regular by [[3], Theorem 3.13].

Remark 2.6. Let T=ℤ2×ℤ2 be a semi-simple ring and s=(1,0)∈T. Then any element a ∈ T satisfies a2=a and 2a=0. Let R=T[x]/〈sx,x2〉with x the indeterminate and S={1,s} be a multiplicative subset of R. By [[3], Example 3.18], R is u-S-von Neumann regular and not von Neumann regular, so there exsits an R-module which is w-u-S-flat but not flat (see, Proposition 2.5).

Recall that an R-module M is said to be an S-torsion-free module if sx=0, for s ∈ S and x ∈ M, implies x=0.

Lemma 2.7. Let R be a ring, S be a multiplicative subset of R, and M be an R-module. If M is a w-u-S-flat, then HomR(M,E) is injective for any injective S-torsion-free R-module E.

Proof. Let I be an ideal of R and E be an injective S-torsion-free. By [[2], Theorem 3.4.11] we have the isomorphism

Since, M is w-u-S-flat, we have that Tor1R(R/I,M) is u-S-torsion and by [[3], Proposition 2.5] we have HomR(Tor1R(R/I,M),E)=0. Thus, ExtR1(R/I,HomR(M,E))=0 which implies that HomR(M,E) is injective.

Proposition 2.8. Let R be a ring, S be a multiplicative subset of R. Then the following statements hold.

• 1. The class of all w-u-S-flat modules is closed under pure submodules and pure quotients.

• 2. Any finite direct sum of w-u-S-flat modules is w-u-S-flat.

• 3. Let 0→A→B→C→0 be a u-S-exact sequence. If A is u-S-torsion. Then B is w-u-S-flat if and only if C is w-u-S-flat.

• 4. Let 0→A→B→C→0 be a u-S-exact sequence. If C is w-u-S-flat with u-S-fdR(C)≤1. Then A is w-u-S-flat if and only if B is w-u-S-flat.

Proof. (1). Let I be an ideal of R. Suppose 0→M→N→L→0 is a pure exact sequence. We have the following commutative diagram with rows exact:

By the S-analogue of the Five Lemma (see[[5], Theorem 1.3]), the natural homomorphism f:M⊗RI→M⊗RR and g:L⊗RI→L⊗RR are all u-S-monomorphisms. Consequently, M and L are all w-u-S-flat by Proposition 2.3.

(2). Let F1,…,Fn be a w-u-S-flat modules and I be an ideal of R. Then, there exists s_i∈ S such that siTor1R(R/I,Fi)=0. Set s=s1…sn. Thus,

which implies that ⊕i=1nFi is w-u-S-flat.

(3). Let 0→A→B→C→0 be a u-S-exact sequence and I be an ideal of R. By [[5], Theorem 1.5], we have the following u-S-exact sequence

Since A is u-S-torsion, we get that Tor1R(R/I,A) and R/I⊗RA are u-S-torsion by [[3], Corollary 2.6]. Hence, Tor1R(R/I,B) u-S-torsion if and only if Tor1R(R/I,C) u-S-torsion, which implies that B is w-u-S-flat if and only if C is w-u-S-flat.

(4). Let 0→A→B→C→0 be a u-S-exact sequence and I be an ideal of R. By [[5], Theorem 1.5], we have the following u-S-exact sequence

The left term is u-S-torsion by [[5], Proposition 2.3] and the right term is u-S-torsion since C is w-u-S-flat. Hence, Tor1R(R/I,A) u-S-torsion if and only if Tor1R(R/I,B) u-S-torsion, which implies that A is w-u-S-flat if and only if B is w-u-S-flat.

Lemma 2.9. Let R be a ring and S a multiplicative subset of R. If A is a flat R-module and B a w-u-S-flat R-module, then, A⊗RB is w-u-S-flat R-module.

Proof. Let I be an ideal of R. By [[2], Theorem 3.4.10] we have the isomorphism

For any s∈ S we have

Since B is a w-u-S-flat, Tor1R(R/I,B) is a u-S-torsion with respect to, say s. So sTor1R(R/I,B)=0. Thus,

Hence, Tor1R(R/I,A⊗RB) is u-S-torsion. Then, A⊗RB is a w-u-S-flat.

Proposition 2.10. Let R be a ring, S be a multiplicative subset of R. If M is w-u-S-flat over a ring R, then M_S is flat over R_S. The converse holds if S consists of finite elements.

Proof. Let I_S be an ideal of R_S, where I is an ideal of R. Then there exists s ∈ S such that sTor1R(R/I,M)=0. Hence, by [[2], Theorem 3.4.12], we have 0=Tor1R(R/I,M)S≅ Tor1RS(RS/IS,MS). So M_S is flat over R_S. For the converse, let I be an ideal of R. By [[2], Theorem 3.4.12] again, we have Tor1R(R/I,M)S=0 which implies that Tor1R(R/I,M) is S-torsion by [[2], Example 1.6.13]. Hence, Tor1R(R/I,M) is u-S-torsion by [[3], Proposition 2.3] and so M is w-u-S-flat.

By Proposition 2.10 and [[3], Proposition 3.8] we have the following corollary.

Corollary 2.11. Let R be a ring, S be a multiplicative subset of R consisting of finite elements. Then, every w-u-S-flat R-module is u-S-flat.

Let p be a prime ideal of R. We say an R-module M is w-u-p-flat shortly provided that M is w-u-(R− p)-flat.

Proposition 2.12. Let R be a ring and M an R-module. Then the following statements are equivalent:

• 1. M is flat.

• 2. M is w-u-p-flat for any p∈Spec(R).

• 3. M is w-u-m-flat for any m∈Max(R).

Proof. (1)⇒(2)⇒(3). These are trivial.

(3)⇒(1). Let I be an ideal of R. Hence, Tor1R(R/I,M) is u-(R−m)-torsion. Then, for any m∈Max(R), there exists sm∈S such that smTor1R(R/I,M)=0. Since the ideal generated by all sm is R, Tor1R(R/I,M)=0. So M is flat.

Let R be a ring and M an R-module. R[x] denotes the polynomial ring with one indeterminate, where all coefficients are in R. Set M[x]=M⊗RR[x], then M[x] can be seen as an R[x]-module naturally.

Proposition 2.13. Let R be a ring, S be a multiplicative subset of R and M is an R[x]-module. If M is w-u-S-flat over R[x], then M is w-u-S-flat over R.

Proof. Suppose that M is a w-u-S-flat R[x]-module. Then it is easy to verify that M[x] is also a w-u-S-flat R[x]-module. By [[1], Theorem 1.3.11], Tor1R(R/I,M)[x]≅Tor1R[x]((R/I)[x],M[x])=Tor1R[x](R[x]/I[x],M[x]) is u-S-torsion. Hence, there exists an element s ∈ S such that sTor1R(R/I,M)[x]=0. Thus, sTor1R(R/I,M)=0. It follows that M is a w-u-S-flat R-module.

Let R be a ring. The flat dimension of an R-module M is defined as the shortest flat resolution of M. In this section, we introduce and investigate the notion of weak u-S-flat dimension of modules and rings as follows.

Defenition 3.1. If M is an R-module, then w-u-S-fdR(M) (w-u-S-fd abbreviates weak u-S-flat dimension) if there is a u-S-exact sequence of R-modules

where each F_i is a u-S-flat (i=0,⋯,n−1) and F_n is w-u-S-flat. The u-S-exact sequence (∗) is called a w-u-S-flat u-S-resolution of length n of M. If no such finite w-u-S-flat u-S-resolution exists, then w-u-S-fdR(M)=∞; otherwise, define w-u-S-fdR(M)=n if n is the length of a shortest w-u-S-flat u-S-resolution of M.

The weak u-S-flat dimension of R is defined by:

Obviously, w-u-S-fdR(M)≤u-S-fdR(M)≤fdR(M), with equality when S is composed of units. However, this inequality may be strict (see, Remarks 2.2 and 2.6). It is also obvious that an R-module M is w-u-S-flat if and only if w-u-S-fdR(M)=0. Also, w-u-S-w.gl.dim(R)≤u-S-w.gl.dim(R)≤w.gl.dim(R), with equality when S is composed of units, and this inequality may be strict (see, Proposition 2.5 and [[3], Example 3.18]).

By Corollary 2.4, we have the following Lemma.

Lemma 3.2. Let R be a ring, S a multiplicative subset of R. If A is u-S-isomorphic to B, then w-u-S-fdR(A)=w-u-S-fdR(B).

In the next result, we give a description of the w-u-S-flat dimension of modules.

Proposition 3.3. Let R be a ring and S be a multiplicative subset of R. The following statements are equivalent for an R-module M.

• 1. w-u-S-fdR(M)⩽n.

• 2. Torn+1R(R/I,M) is u-S-torsion for any ideal I of R.

• 3. Torn+1R(R/I,M) is u-S-torsion for any finitely genrated ideal I of R.

• 4. If the sequence 0→Fn→Fn−1→⋯→F0→M→0 is an exact with F0,⋯,Fn−1 are flat R-modules, then F_n is w-u-S-flat.

• 5. If the sequence 0→Fn→Fn−1→⋯→F0→M→0 is a u-S-exact with F0,⋯,Fn−1 are u-S-flat R-modules, then F_n is w-u-S-flat.

• 6. If the sequence 0→Fn→Fn−1→⋯→F0→M→0 is an exact with F0,⋯,Fn−1 are u-S-flat R-modules, then F_n is w-u-S-flat.

• 7. If the sequence 0→Fn→Fn−1→⋯→F0→M→0 is a u-S-exact with F0,⋯,Fn−1 are flat R-modules, then F_n is w-u-S-flat.

• 8. There exists a u-S-exact sequence 0→Fn→Fn−1→⋯→F0→M→0, where F0,⋯,Fn−1 are flat R-modules and F_n is w-u-S-flat.

• 9. There exists an exact sequence 0→Fn→Fn−1→⋯→F0→M→0, where F0,⋯,Fn−1 are flat R-modules and F_n is w-u-S-flat.

• 10. There exists an exact sequence 0→Fn→Fn−1→⋯→F0→M→0, where F0,⋯,Fn are w-u-S-flat.

Proof. (1)⇒(2). We prove (2) by induction on n. For the case n=0, (2) holds by Proposition 2.3 as M is a w-u-S-flat module. If n>0, then there is a u-S-exact sequence 0→Fn→Fn−1→⋯→F0→M→0 with all F_i u-S-flat (i=0,⋯,n−1) and F_n is w-u-S-flat. Let K0=ker(F0→M). We have two u-S-exact sequences 0→K0→F0→M→0 and 0→Fn→Fn−1→…→F1→K0→0. We note that w-u-S-fdR(K0)⩽n−1. Hence, by induction we have, TornR(R/I,K0) is u-S-torsion for any ideal I of R. Thus, it follows from [[5], Corollary 1.6], that TornR(R/I,M)) is u-S-torsion.

(2)⇒(3). This is obvious.

(3)⇒(4). Let 0→Fn→Fn−1→⋯→F0→M→0 be an exact sequence. Set K0=ker(F0→M) and Ki=ker(Fi→Fi−1), where (i=1,…,n−1). Since all F0,F1,…,Fi−1 are flat, Tor1R(R/I,Fn)≅Torn+1R(R/I,M) is u-S-torsion for all finitely generated ideal I of R. Thus, F_n is a w-u-S-flat module by Proposition 2.3.

(4)⇒(1). Trivial.

(3)⇒(5). Let 0→Fn→Fn−1→⋯→F0→M→0 be a u-S-exact sequence. Set Ln=Fn and Li=Ima(Fi→Fi−1), where (i=1,…,n−1). Then both 0→Li+1→Fi→Li→0 and 0→L1→F0→M→0 are u-S-exact sequences.

By using [[5], Corollary 1.6] repeatedly, we can obtain that Tor1R(Fn,R/I) is u-S-torsion for all finitely generated ideal I of R, which implies that F_n is w-u-S-flat by Proposition 2.3.

(5)⇒(6)⇒(4) and (5)⇒(7)⇒(4). These implications are trivial.

(4)⇒(9). Let …→Pn→Pn−1→fPn−2…→P0→M→0 be a projectiveresolution of M. Set Fn=Ker(f). Then we have an exact sequence 0→Fn→Pn−1→fPn−2…→P0→M→0. By (4), F_n is w-u-S-flat. So (9) holds.

(9)⇒(10)⇒(1) and (9)⇒(8)⇒(1). These are obvious.

Corollary 3.4. Let R be a ring and S′⊆S multiplicative subsets of R. Suppose M is an R-module, then w-u-S-fdR(M)≤w-u-S′-fdR(M).

Proof. Suppose S′⊆S are multiplicative subsets of R. Let M be an R-modules and I be an ideal of R. If Torn+1R(R/I,M) is u-S'-torsion, then Torn+1R(R/I,M) is u-S-torsion. Hence, by Proposition 3.3., we have the result.

Corollary 3.5. Let R be a ring, S a multiplicative subset of R and M an R-module. Then, fdRS(MS)≤w-u-S-fdR(M). Moreover, if S is composed of finite elements, then w-u-S-fdR(M)=fdRS(MS).

Proof. Let 0→Fn→Fn−1→…→F1→F0→M→0 be an exact sequence, where F0,F1,…,Fn−1 are flat R-modules. By localizing at S, we get an exact sequence of R_S-modules, 0→(Fn)S→(Fn−1)S→…→(F1)S→(F0)S→(M)S→0. By Proposition 2.10, if F_n is w-u-S-flat, so (Fn)S is flat over R_S, and the converse if S composed of finite elements. Hence, the desired result follows.

The proof of the next proposition is standard homological algebra. Thus we omit its proof.

Proposition 3.6. Let R be a ring, S be a multiplicative subset of R, and 0→M″→M′→M→0 be an exact sequence of R-modules. If two of w-u-S-fdR(M″), w-u-S-fdR(M′) and w-u-S-fdR(M) are finite, so is the third. Moreover

• 1. w-u-S-fdR(M″)≤maxw-u-S-fdR(M′), w-u-S-fdR(M)−1.

• 2. w-u-S-fdR(M′)≤max{w-u-S-fdR(M″), w-u-S-fdR(M)}.

• 3. w-u-S-fdR(M)≤max{w-u-S-fdR(M′), w-u-S-fdR(M″)+1}.

Corollary 3.7. Let R be a ring, S be a multiplicative subset of R, and 0→M″→M′→M→0 be an exact sequence of R-modules. If M' is w-u-S-flat and w-u-S-fdR(M)>0, then w-u-S-fdR(M)=w-u-S-fdR(M″)+1.

Proposition 3.8. Let R be a ring, S be a multiplicative subset of R, and {Mi} be a finite family of R-modules. Then w-u-S-fdR(⊕iMi)=supi{w-u-S-fdR(Mi)}.

Proof. The proof is straightforward.

Proposition 3.9. Let R be a ring, S be a multiplicative subset of R, and n≥ 0 be a an integer. Then the following statements are equivalent:

• 1. w-u-S-w.gl.dim(R)≤n.

• 2. w-u-S-fdR(M)⩽n for all R-modules M.

• 3. w-u-S-fdR(R/J)⩽n for all ideals J of R.

• 4. Torn+1R(R/I,M) is u-S-torsion for any R-module M and any ideal I of R.

• 5. Torn+1R(R/I,M) is u-S-torsion for any R-module M and any finitely generated ideal I of R.

Consequently, we have

Proof. (1)⇔(2)⇒(3) and (4)⇒(5). The are obvious.

(2)⇒(4) and (5)⇒(2). These are immediate from Proposition 3.3.

(3)⇒(1). Let J be an ideal of R, so w-u-S-fdR(R/J)⩽n by (3). By Proposition 3.3, Torn+1R(R/I,R/J) is u-S-torsion for any ideal I of R. Thus, there exists s∈ S such that sTorn+1R(R/I,R/J)=0 and so by [[5], Proposition 3.2], we have u-S-w.gl.dim(R)≤n for any R-module M. Thus, w-u-S-w.gl.dim(R)≤n.

Next, we show that rings R with w-u-S-w.gl.dim(R)=0 are exactly u-S-von Neumann regular rings.

Proposition 3.10. Let R be a ring, S be a multiplicative subset of R. The following are equivalent:

• 1. w-u-S-w.gl.dim(R)=0.

• 2. Every R-module is w-u-S-flat.

• 3. R/I is w-u-S-flat for any ideal I of R.

• 4. R is a u-S-von Neumann regular ring.

Proof. The equivalence of (1), (2), and (3), follows from Proposition 3.9.

(2)⇔(4). Follows from Proposition 2.5.

The proof of the follwing Proposition fllows from Proposition 3.9. Thus, we omit its proof.

Proposition 3.11. Let R be a ring, S be a multiplicative subset of R. Then the following are equivalent:

• 1. w-u-S-w.gl.dim(R)≤1.

• 2. Every submodule of w-u-S-flat R-module is w-u-S-flat.

• 3. Every submodule of flat R-module is w-u-S-flat.

• 4. Every ideal of R is w-u-S-flat.

Let θ:R→T be a ring homomorphism. Suppose S is a multiplicative subset of R, then θ(S)={θ(s)|s∈S} is a multiplicative subset of T.

Lemma 3.12. Let θ:R→T be a ring homomorphism, S a multiplicative subset of R. Suppose L is a w-u-θ(S)-flat T-module. Then for any ideal I of R and any n≥ 0,TornR(R/I,L) is u-S-isomorphic to TornR(R/I,T)⊗TL. Consequently, w-u-S-fdR(L)≤w-u-S-fdR(T).

Proof. Similar to proof [[5], Lemma 4.1].

Proposition 3.13. Let θ:R→T be a ring homomorphism, S a multiplicative subset of R. Suppose M is an T-module. Then

Proof. Suppose that w-u-θ(S)-fdT(M)=n<∞. If n=0, then M is w-u-θ(S)-flat over T. By Lemma 3.12, w-u-S-fdR(M)≤n+w-u-S-fdR(T).

Now we assume n>0. Let 0→A→F→M→0 be an exact sequence of T-modules, where F is a free T-module. Then w-u-θ(S)-fdT(A)=n−1 by Corollary 3.7. By induction, w-u-S-fdR(A)≤n−1+w-u-S-fdR(T). Note that w-u-S-fdR(T)=w-u-S-fdR(F). By Proposition 3.6, we have

Proposition 3.14.Let R be a ring, S a multiplicative subset of R and M an R-module. Then, w-u-S-fdR[x](M[x])=w-u-S-fdR(M).

Proof. Suppose that w-u-S-fdR(M)≤n. Then Torn+1R(R/I,M) is u-S-torsion for any ideal I of R. Let I[x] be an ideal of R[x]. By [[1], Theorem 1.3.11], we have Torn+1R[x]((R/I)[x],M[x])≅Torn+1R(R/I,M)⊗RR[x]. And by [[3], Corollary 2.6], we have Torn+1R(R/I,M)⊗RR[x] is u-S-torsion since Torn+1R(R/I,M) is u-S-torsion. Thus, Torn+1R[x]((R/J)[x],M[x]) is u-S-torsion, which implies that, w-u-S-fdR[x](M[x])≤n by Proposition 3.3.

Conversely, Let 0→Fn→…→F1→F0→M[x]→0 be an exact sequence with each F_i u-S-flat over R[x](1≤i≤n−1) and F_n w-u-S-flat over R[x]. Hence, it is also a w-u-S-flat resolution of M[x] over R by Proposition 2.13. Then, by Proposition 3.3, we have Torn+1R(R/I,M[x]) is u-S-torsion for any ideal I of R. It follows that sTorn+1R(R/I,M[x])=s⊕n=1∞ Torn+1R(R/I,M)=0. Hence, Torn+1R(R/I,M) is u-S-torsion. Consequently, w-u-S-fdR(M)≤w-u-S-fdR[x](M[x]) by Proposition 3.3 again.