HIGHER DERIVATIONS ON RINGS AND MODULES PAUL E. BLAND Received 27 June 2001 and in revised form 14 July 2005

PAUL E. BLAND

Received 27 June 2001 and in revised form 14 July 2005

Letτ be a hereditary torsion theory on Mod_R and suppose thatQ_τ: Mod_R→Mod_R is the localization functor. It is shown that for allR-modulesM, every higher derivation defined onMcan be extended uniquely to a higher derivation defined onQτ(M) if and only ifτis a higher differential torsion theory. It is also shown that ifτis a TTF theory andC_τ:M→Mis the colocalization functor, then a higher derivation defined onMcan be lifted uniquely to a higher derivation defined onCτ(M).

1. Introduction

Rim has shown in [16] that under certain conditions a higher antiderivationd:M→M can be extended to a higher antiderivation d^τ:Qτ(M)→Qτ(M), where Qτ: Mod_R→ Mod_R is the localization functor [10] at a hereditary torsion theoryτon Mod_R. By se- lecting the involution on the ring in the definition of a higher antiderivation to be the identity mapping onR, a higher antiderivationd:M→Mbecomes a higher derivation as defined by Ribenboim in [15]. Thus Rim’s results, which generalize the results of Golan [9], show that a higher derivationd:M→M can be extended to a higher derivation d^τ:Qτ(M)→Qτ(M) whenever the conditions of his proposition are met. Uniqueness of extensions of higher (anti-) derivations and the necessary and sufficient conditions for the existence of these extensions were not addressed. The purpose of this paper is to intro- duce higher differential torsion theories and to show that a higher derivationd:M→M can always be extended uniquely toQτ(M), the module of quotients ofM, if and only if τis a higher differential torsion theory on ModR. We also show that a higher derivation d:M→M can be lifted uniquely to the module of coquotientsCτ(M) ofM at a TTF theory on Mod_R.

Throughout this paperRwill be an associative ring with identity 1, Mod_Rwill denote the category of unitary rightR-modules and all modules and module homomorphisms will be in Mod_R.

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:15 (2005) 2373–2387 DOI:10.1155/IJMMS.2005.2373

2. Differential torsion theory

A torsion theoryτon Mod_Ris a pair (T, F) of classes ofR-modules such that the following conditions hold.

(1) T∩F=0.

(2) IfM→N→0 is an exact sequence in Mod_RandM∈T, thenN∈T.

(3) If 0→M→Nis an exact sequence in Mod_RandN∈F, thenM∈F.

(4) For eachR-moduleM, there is a short exact sequence 0→T→M→F→0 in Mod_RwithT∈T andF∈F.

It follows that the class T is closed under factor modules, direct sums, and extensions, and that F is closed under submodules, direct products, and extensions. A class C ofR- modules is said to beclosed under extensionsif whenever 0→M1→M→M2→0 is a short exact sequence in Mod_R andM1andM2are in C, thenM is in C. Modules in T will be calledτ-torsionand those in F are calledτ-torsion-free. EachR-moduleMhas a largest and necessarily uniqueτ-torsion submodule given bytτ(M)=

N∈SN, where S is the set ofτ-torsion submodules ofM. A torsion theory will be calledhereditaryif T is closed under submodules and it will be calledcohereditaryif F is closed under factor modules.

Standard results and terminology on torsion theory can be found in [4,10] while general information on rings and modules can be found in [2]. Finally, ifN is a submodule of anR-moduleM, then for anyx∈M, (N:x) will denote the right ideal of Rgiven by {a∈R|xa∈N}and (0 :x) is the right ideal{a∈R|xa=0}.

A nonempty collectionᏲof right ideals ofRis said to be a (Gabriel)filter[8] if the following two conditions hold.

(1) IfK∈Ᏺ, then (K:a)∈Ᏺfor eacha∈R.

(2) IfI is a right ideal ofRandK∈Ᏺis such that (I:a)∈Ᏺfor eacha∈K, then I∈Ᏺ.

It can be shown that each filter of right ideals ofRalso satisfies the following three conditions.

(3) IfJ∈ᏲandKis a right ideal ofRsuch thatJ⊆K, thenK∈Ᏺ. (4) IfJ,K∈Ᏺ, thenJ∩K∈Ᏺ.

(5) IfJ,K∈Ᏺ, thenJK∈Ᏺ.

Ifτis a hereditary torsion theory on Mod_R, thenᏲτ= {K|Kis a right ideal ofRand R/K∈T}is a filter. An elementxof anR-moduleM is said to be aτ-torsion elementof Mif there is aK∈Ᏺτsuch thatxK=0. The set of allτ-torsion elements ofMis theτ- torsion submodulet_τ(M) ofMmentioned earlier. Moreover, anR-moduleMisτ-torsion iftτ(M)=Mandτ-torsion-free iftτ(M)=0. Conversely, ifᏲis a filter of right ideals of Randt(M)= {x∈M|xK=0 for someK∈Ᏺ}, thenτ=(T, F) is a hereditary torsion theory on Mod_R, where T= {M|t(M)=M}and F= {M|t(M)=0}. It follows that there is a one-to-one correspondence between hereditary torsion theories on Mod_Rand filters of right ideals ofR.

An additive mappingδ:R→Rsuch thatδ(ab)=δ(a)b+aδ(b) for alla,b∈Ris said to be aderivation onRand, given a derivationδ onR, an additive mappingd:M→M such thatd(xa)=d(x)a+xδ(a) is aderivation onM. Important to our discussion is the concept of a differential torsion theory, introduced in [5]. Ifδis a derivation onRand Ᏺis a filter of right ideals ofR, thenᏲis called adifferential filter(with respect toδ) if

for eachK∈Ᏺthere is anI∈Ᏺsuch thatδ(I)⊆K. Ifτis a hereditary torsion theory on Mod_R and the corresponding filterᏲτ is a differential filter, thenτ is said to be a differential torsion theory.

Remark 2.1. IfᏲis a differential filter andK∈Ᏺ, then there is anI∈Ᏺsuch thatδ(I)⊆ K. The right idealI can actually be chosen so thatI⊆K. Indeed, ifI∈Ᏺis such that δ(I)⊆KandI=I∩K, thenI∈Ᏺ,I⊆K, andδ(I)⊆K.

The following example shows that differential torsion theories do indeed exist. We will see additional examples later.

Example 2.2. If S is a multiplicatively closed set of elements of R that is a right denominator set[12], thenSsatisfies the following conditions.

(1) If (a,s)∈R×S, then there is a (b,t)∈R×Ssuch thatat=sb.

(2) Ifsa=0 withs∈Sanda∈R, thenat=0 for somet∈S.

Ifδ is a derivation on R, then the setᏲ= {K|Kis a right ideal ofRandK∩S= ∅}

is a filter of right ideals ofR. IfK∈Ᏺ, lets∈K∩S. Since (δ(s),s)∈R×S, there is a (b,t)∈R×Ssuch thatδ(s)t=sb. Nowδ(st)=δ(s)t+sδ(t)=sb+sδ(t)∈sR⊆K, so if a∈R, thenδ(sta)=δ(st)a+stδ(a)∈K. Henceδ(stR)⊆K. ThereforeᏲis a differential filter, so the torsion theory determined byᏲis a differential torsion theory.

The following lemma gives two conditions that characterize differential filters.

Lemma2.3. Letτbe a hereditary torsion theory onMod_Rwith corresponding filterᏲτ. Then the following are equivalent for a derivationδonR.

(1)Ᏺτis a differential filter.

(2)For everyR-moduleM, ifx∈tτ(M), then there is anI∈Ᏺτsuch thatδ(I)⊆(0 :x).

(3)For everyR-moduleM, ifd:M→Mis a derivation onM, thend(tτ(M))⊆tτ(M).

Proof. (1)⇒(3). If x∈tτ(M), then there is aK∈Ᏺτ such thatxK=0 and anI∈Ᏺτ

such thatδ(I)⊆K. Thus ifa∈I∩K∈Ᏺτ, then we see that 0=d(xa)=d(x)a+xδ(a)= d(x)a. Henced(x)(I∩K)=0, sod(x)∈tτ(M).

(3)⇒(2). Ifx∈tτ(M) anda∈R, thend(x) andd(xa) are intτ(M). Thus (0 :d(x))∩ (0 :d(xa))∈ Ᏺτ. Therefore I =(0 :d(x))∩(0 :d(xa))∈Ᏺτ. If a∈I, then d(x)a= d(xa)=0, so 0=d(xa)=d(x)a+xδ(a)=xδ(a). Thusδ(a)∈(0 :x) and we haveδ(I)⊆ (0 :x).

(2)⇒(1). IfK∈Ᏺτ, then we need to find anI∈Ᏺτsuch thatδ(I)⊆K. SinceR/K is τ-torsion, 1 +K∈t_τ(R/K), so by (2) there is anI∈Ᏺτ such thatδ(I)⊆(0 : 1 +K)=K.

3. Higher differential torsion theory

Letnbe a nonnegative integer. Then a family of additive mappingδ= {δi:R→R}ⁿ_i=0, denoted byδ:R→R, is said to be ahigher derivation(onR) of ordernprovided that

δi(ab)=δi(a)δ0(b) +δi−1(a)δ1(b) +···+δ0(a)δi(b)

=δi(a)b+δi−1(a)δ1(b) +···+aδi(b) (3.1)

for alla,b∈Randi=0, 1,...,n, where it is understood thatδ0is the identity mapping on R. Ifδ:R→Ris a higher derivation of order 1, thenδ1(ab)=δ1(a)b+aδ1(b) for alla,b∈ R, so a higher derivation onRof order 1 gives a derivation onR. Given a higher derivation δ:R→Rof order n, a family of additive mappings d= {di:M→M}ⁿ_i=0, denoted by d:M→M, is ahigher derivation(onM) of ordernif forx∈Manda∈R

di(xa)=di(x)δ0(a) +di−1(x)δ1(a) +···+d0(x)δi(a)

=di(x)a+di−1(x)δ1(a) +···+xδi(a) (3.2) fori=0, 1,...,nwith the understanding thatd0will always be the identity mapping onM.

A higher derivationd:M→Mof order 1 givesd1(xa)=d1(x)a+xδ1(a), sodproduces a derivation onM. To simplify terminology, a higher derivation of ordernwill be referred to simply as a derivation of ordern.

Remark 3.1. Ifδ:R→Randd:M→Mare derivations of ordernand 0≤k≤n, then ¯δ= {δi:R→R}^k_i=0and ¯d= {di:M→M}^k_i=0are derivations of orderk, where ¯dis taken with respect to ¯δ. This will subsequently be described by saying thatdproduces derivations of orderkfork=0, 1,...,n.

LetᏲbe the filter of right ideals ofRand suppose thatδ:R→Ris derivation of order n. IfᏲis such that for eachK∈Ᏺthere is anI∈Ᏺsuch thatδi(I)⊆Kfori=0, 1,...,n, then we will say thatᏲis adifferential filter of ordern. In this setting, the corresponding hereditary torsion theoryτis called adifferential torsion theory of ordern. We now fix the derivationδ:R→Rof ordernand assume, unless stated otherwise, that every derivation d:M→Mof ordern, every differential filterᏲof ordern, and every differential torsion theoryτof ordernis taken with respect toδ. Because ofRemark 3.1every differential filter (differential torsion theory) of ordernproduces a differential filter (differential tor- sion theory) of orderkfork=0, 1,...,n.

In the previous section an example of a differential torsion theory was given. Differ- ential torsion theories of ordern can also be shown to exist. In each of the following examplesδ is a derivation onR of ordern. In particular, we also see that each of the following is also a differential torsion theory.

Example 3.2. LetRbe a commutative ring and suppose thatᏲis a filter of ideals ofR. If I∈Ᏺ, thenI²∈Ᏺ, and it follows thatδ0(I²)=I²⊆I. So suppose thatδi(I²)⊆Ifor alli with 0≤i < n. Ifa,b∈I, then

δn(ab)=δn(a)b+δn−1(a)δ1(b) +···+δ1(a)δn−1(b) +aδn(b) (3.3) which is easily seen to be inI. Sinceδ_nis additive, we have thatδ_n(I²)⊆I. ThereforeᏲτ

is a differential filter of ordern, so the corresponding hereditary torsion theory on ModR

is a differential torsion theory of ordern. Thus for a commutative ring every hereditary torsion theory on Mod_Ris a differential torsion theory of ordern.

Example 3.3. Jans has shown in [11] that ifτ=(T, F) is a hereditary torsion on Mod_R such that T is closed under direct products, then there is an idempotent idealI∈Ᏺτsuch

thatI⊆Kfor eachK∈Ᏺτ. Using the same procedure as in the previous example, we can show thatδ_i(I)⊆Kfori=0, 1,...,n. Thusτis a differential torsion theory of ordern.

Example 3.4. IfRis left perfect, then Alin and Armendariz [1] and Dlab [7] have indepen- dently proved that ifτ=(T, F) is a hereditary torsion theory on Mod_R, then T is closed under direct products. Thus, we see from the previous example that ifRis left perfect, then every hereditary torsion theory on Mod_Ris a differential torsion theory of ordern.

With the definitions of a differential filter of ordernand a differential torsion the- ory of ordernin place, one might expect thatLemma 2.3can be generalized to higher differential filters. The following lemma shows that this is indeed the case.

Lemma3.5. Ifτis a hereditary torsion theory onMod_Rwith corresponding filterᏲτ, then the following are equivalent for an integern≥0.

(1)Ᏺτis a differential filter of ordern.

(2)For everyR-moduleMand each derivationd:M→Mof ordern, ifx∈tτ(M), then there is anI∈Ᏺτsuch thatδ_i(I)⊆(0 :d_n−i(x))fori=0, 1,...,n.

(3)For everyR-moduleM, ifd:M→M is a derivation of ordern, thendi(tτ(M))⊆ tτ(M)fori=0, 1,...,n.

Proof. (1)⇒(3). If n=0, then it is trivial that (1)⇒(3), so suppose (1)⇒(3) for every integerk, 0≤k < n. Now letᏲτbe a differential filter of ordernand suppose thatd:M→ M is a derivation of ordern. In view ofRemark 3.1,dproduces a derivation{di:M→ M}ⁿ_i=⁻0¹onMof ordern−1 and it follows thatᏲτis a differential filter of ordern−1. So ifx∈t_τ(M), then the induction hypothesis shows thatd1(x),...,d_n−1(x)∈t_τ(M). Hence each of

(0 :x),0 :d1(x),0 :d2(x),...,0 :dn−1(x) (3.4) is inᏲτ. SinceᏲτis a filter,K= ∩ⁿ_i=⁻0¹(0 :d_i(x))∈Ᏺτ, so sinceᏲτis a differential filter of ordern, there is anI∈Ᏺτsuch thatδi(I)⊆Kfori=0, 1,...,n. Ifa∈I, then

d_n(xa)=d_n−1(x)δ1(a)= ··· =d1(x)δ_n−1(a)=xδ_n(a)=0, (3.5) so

dn(xa)=dn(x)a+dn−1(x)δ1(a) +···+d1(x)δn−1(a) +xδn(a) (3.6) givesd_n(x)a=0. Therefore,d_n(x)I=0, which indicates thatd_n(x)∈t_τ(M) and we have (3).

(3)⇒(2). Ifn=0, then (3)⇒(2) is trivial since we can letI=(0 :x). Now suppose that (3)⇒(2) for every integerk, 0≤k < n, and let (3) hold for n. Ifd:M→M is a derivation of ordernandx∈t_τ(M), then since (3) holds forn, we haved_i(x)∈t_τ(M) for i=0, 1,...,n. Since{di:M→M}ⁿ_i=⁻0¹is a derivation onM of ordern−1, the induction hypothesis givesI∈Ᏺτsuch thatδ_i(I)⊆(0 :d_n−1−i(x)) fori=0, 1,...,n−1. Ifa∈I= I∩(0 :d_n(x))∈Ᏺτ, then

dn(xa)=dn(x)a=dn−1(x)δ1(a)=dn−2(x)δ2(a)= ··· =d1(x)δn−1(a)=0, (3.7)

so

d_n(xa)=d_n(x)a+d_n−1(x)δ1(a) +···+d1(x)δ_n−1(a) +xδ_n(a) (3.8) becomes

xδn(a)=0. (3.9)

Henceδ_n(I)⊆(0 :x), soδ_i(I)⊆(0 :d_n−i(x)) fori=0, 1,...,n.

(2)⇒(1). It is obvious that (2)⇒(1) whenn=0, so suppose that (2)⇒(1) for every inte- gerksuch that 0≤k < n. IfK∈Ᏺτ, thenR/Kisτ-torsion, so letd= {di:R/K→R/K}^k_i=0, be a derivation onR/Kof orderkfork=0, 1,...,n. Since (2) holds fork=0, 1,...,n−1, for each suchkand eachx+K∈R/Kthere is anI_k,x∈Ᏺτsuch thatδ_i(I_k,x)⊆(0 :d_k−i(x+ K)) fori=0, 1,...,k. In particular, for eachx+K∈R/K, we haveδk(Ik,x)⊆(0 :x+K) for k=0, 1,...,n−1. Ifa∈Ix=[∩ⁿ_k=⁻¹0Ik,x]∩(0 :dn(x+K))∈Ᏺτ, then

0=dn

(x+K)a=dn(x+K)a+dn−1(x+K)δ1(a) +···+ (x+K)δn(a)=(x+K)δn(a).

(3.10) Henceδk(Ix)⊆(0 :x+K) fork=0, 1,...,n. Ifx=1, this givesδk(I1)⊆(0 : 1 +K)=Kfor eachk, soᏲτis a differential filter of ordern.

4. Higher derivations and modules of quotients

If τ is a torsion theory on Mod_R, then an R-module Q_τ(M) together with an R- homomorphismϕ:M→Qτ(M) is said to be alocalizationofMatτprovided that kerϕ and cokerϕareτ-torsion andQ_τ(M) isτ-injective andτ-torsion-free. AnR-moduleMis said to beτ-injectiveif Hom_R(−,M)=0 preserves short exact sequences 0→N1→N→ N2→0, whereN2is aτ-torsion module. The moduleQτ(M), called themodule of quo- tientsofM, is unique up to isomorphism whenever it can be shown to exist. Ohtake [14]

has shown that a localizationϕ:M→Q_τ(M) exists for everyR-moduleMif and only if the torsion theory is hereditary. It is well known that ifτis hereditary, then we can set Qτ(M)=Eτ(M/tτ(M)), whereEτ(M/tτ(M)) is theτ-injective envelopeofM/tτ(M) [4,10].

Ifη:M→M/t_τ(M) is the natural surjection andµ:M/t_τ(M)→Q_τ(M) is the canonical injection, thenϕ=µη.

Ifd:M→Mis a derivation of ordern, then we say thatdcan be extended to a deriva- tiond^τ:Q_τ(M)→Q_τ(M) of ordernif the diagram

M ^ϕ

di

Qτ(M)

d^τi

M ^ϕ Q_τ(M)

(4.1)

is commutative fori=0, 1,...,n. We can now show that a derivationd:M→M of or- dernhas a unique extension to a derivationd^τ:Qτ(M)→Qτ(M) of ordernfor every R-moduleM if and only ifτis a differential torsion theory of ordern. This result is es- tablished by the following lemma and proposition.

Lemma4.1. Let τbe a differential torsion theory of ordernonMod_R. If a derivationd: M→Mof orderncan be extended to a derivationd^τ:Q_τ(M)→Q_τ(M)of ordern, thend^τ is unique.

Proof. LetᏲτbe the differential filter of orderncorresponding toτand suppose thatd^τ and ¯d^τare derivations of ordernthat extenddtoQτ(M). Then fork, 0≤k≤n,d^τ, ¯d^τ, anddproduce derivationsd^∗τ, ¯d^∗τ, andd^∗of orderkandd^∗τand ¯d^∗τliftd^∗. SinceᏲτis a differential filter of orderk, ifx∈Qτ(M), then byLemma 3.5there areI,I∈Ᏺτsuch thatδi(I)⊆(0 :d^τ_k−i(x)) andδi(I)⊆(0 : ¯dk−i(x)) fori=0, 1,...,k. Ifa∈K=I∩I∈Ᏺτ, then for eachkwe see that

0=

d^τ_k−d¯^τ_k(xa)

=d_k^τ(x)a+d_k^τ₋₁(x)δ1(a) +···+xδ_k(a)

−d¯_k^τ(x)a−d¯^τ_k−1(x)δ1(a)− ··· −xδk(a)

=

d^τ_k(x)−d¯^τ_k(x)a+d^τ_k−1(x)−d¯_k^τ₋1(x)δ1(a) +···+d^τ1(x)−d¯1^τ(x)δ_i−1(a) + [x−x]δ_k(a)

=

d^τ_k(x)−d¯k(x)a.

(4.2)

Hence, [d^τ_k(x)−d¯^τ_k(x)]K=0, so d_k^τ(x)−d¯_k^τ(x)∈tτ(Qτ(M))=0. Therefored_k^τ=d¯^τ_k for

k=0, 1,...,nand sod^τ=d¯^τ.

We can now establish the main result of this section.

Proposition4.2. Letτbe a hereditary torsion theory onMod_R. Then for everyR-module M, each derivation d:M→M of ordern can be extended uniquely to a derivationd^τ: Qτ(M)→Qτ(M)of ordernif and only ifτis a differential torsion theory of ordern.

Proof. Suppose thatd:M→M is a derivation of ordern. If τis a differential torsion theory of ordern, thenᏲτis a differential filter of ordern, so it follows fromLemma 3.5 that d_i(t_τ(M))⊆t_τ(M) fori=0, 1,...,n. Hencedcan be extended to a derivationd^τ: Qτ(M)→Qτ(M) of ordernsince Rim proved in [16] that such an extension exists when di(tτ(M))⊆tτ(M) fori=0, 1,...,n. Uniqueness follows fromLemma 4.1.

Conversely, suppose that every derivation d:M→M of order n can be extended uniquely to a derivationd^τ:Qτ(M)→Qτ(M) of ordern. From the commutative diagram

M ^ϕ

di

Qτ(M)

d^τi

M ^ϕ Qτ(M)

(4.3)

we see thatϕdi=d_i^τϕfori=0, 1,...,n. So ifx∈tτ(M)=kerϕ, thenϕdi(x)=0 for each i. This gives di(x)∈kerϕ=tτ(M) and so we havedi(tτ(M))⊆tτ(M) fori=0, 1,...,n.

Calling onLemma 3.5again, we see thatτis a differential torsion theory of ordern.

Corollary4.3 [5, Proposition 2.3]. Ifτis a hereditary torsion theory onMod_R, then for everyR-moduleM, each derivationd:M→M can be extended uniquely to a derivation

d^τ:Q_τ(M)→Q_τ(M)if and only ifτ is a differential torsion theory. In particular, τ is a differential torsion theory, then the derivationδ:R→Rextends uniquely to a derivation δ^τ:Qτ(R)→Qτ(R)defined on the ring of the quotients ofR.

Proof. The first part of the corollary is clear if we considerd:M→M to be derivation of order 1. Now letM=Rand apply this result to the derivationδ:R→Rto prove the

second part of the corollary.

One consequence ofProposition 4.2is that for a hereditary torsion theoryτon Mod_R, the right ideals of the filterᏲτform a test set for determining if derivations onMof order ncan be extended uniquely to derivations onQτ(M) of ordern.

5. Higher derivations and modules of coquotients

We now show that a result similar toProposition 4.2holds for colocalizations of modules whenever they universally exist. Colocalizations have been investigated under various ap- proaches by several authors, for example, see [3,6,13].

AnR-moduleCτ(M) together with anR-linear mappingϕ:Cτ(M)→Mis said to be acolocalizationofM atτprovided that kerϕand cokerϕareτ-torsion-free andCτ(M) isτ-torsion andτ-projective. We callC_τ(M) themodule of coquotientsofM. We point out thatτis not assumed to be hereditary. When this is the case, a nonzero submodule of aτ-torsion module can beτ-torsion-free, a condition that is only possible for the zero submodule whenτis hereditary.

AnR-moduleM isτ-projectiveif Hom_R(M,−) preserves short exact sequences 0→ N1→N→N2→0, whereN1is aτ-torsion-free module. Ohtake was also able to show in [14] that a torsion theoryτ is cohereditary if and only if everyR-moduleM has a colocalization atτ. Ifϕ:C_τ(M)→M is a colocalization ofM atτ, then there is anR- epimorphismπ:Cτ(M)→tτ(M) such that ifµ:tτ(M)→M is the canonical injection, thenϕ=µπ. Furthermore, a module of coquotients is unique up to isomorphism when- ever it can be shown to exist.

Ifd:M→Mis a derivation of ordern, then we say thatdcan be lifted to a derivation d^τ:Cτ(M)→Cτ(M) of ordernif the diagram

C_τ(M) ^ϕ

d^τi

M

di

Cτ(M) ^ϕ M

(5.1)

is commutative fori=0, 1,...,n. We will now show that such liftings are always possible at a TTF theoryτon Mod_R.

Whenτ=(T, F) is cohereditary, the class F ofτis both a torsion and a torsion-free class, and the class F generates a hereditary torsion theoryσ=(F, D) on Mod_R. The pair (τ,σ) is often referred to as a TTF theory. Jans has shown in [11] that there is a one- to-one correspondence between TTF theories and idempotent idealsI of R. If (τ,σ) is a TTF theory with corresponding idempotent idealI, then the filter determined byσ is

given byᏲσ= {K⊆R|K⊇I,Ka right ideal ofR}. In this setting,t_τ(R)=Iandt_τ(M)= MI for eachR-moduleM. We have seen inExample 3.3 thatσ is a higher differential torsion theory although this condition onσ is not a factor in lifting higher derivations d:M→Mto higher derivationsd^τ:Cτ(M)→Cτ(M). Sato has shown in [17] that ifτis cohereditary, thenI⊗_RI→^π I→^µ Ris a colocalization ofR, where the mapπ:I⊗_RI→I is given byⁿ_i=1(ai⊗bi)→_n

i=1aibi. Furthermore,I⊗RI is a ring, possibly without an identity, and an (R,R)-bimodule. Sato also shows in [17] thatM⊗RI⊗RI→^π MI→^µ M is a colocalization ofMatτ, where the mapπ:M⊗_RI⊗_RI→MIis given byⁿ_i−1(x_i⊗a_i⊗ bi)→_n

i=1xiaibi. SinceI is an idempotent ideal,δi(I)⊆Ifori=0, 1,...,nand it follows that each derivationd:M→M of ordern is such thatdi(MI)⊆MI fori=0, 1,...,n.

Hence,drestricted toMIproduces a derivationd:MI→MIof ordernthat will also be denoted byd.

We now need the following lemma.

Lemma5.1. IfIis an idempotent ideal ofRandd:M→Mis a derivation of ordern, then the mapρ_i:M×I×I→M⊗_RI⊗_RIgiven by

ρi(x,a,b)=ⁱ

j=0

di−j(x)⊗ _j

k=0

δj−k(a)⊗δk(b)

(5.2) isR-balanced fori=0, 1,...,n. That is,ρiis additive in each variable and such thatρi(xr,a,b)

=ρ_i(x,ra,b)andρ_i(x,ar,b)=ρ_i(x,a,rb)for(x,a,b)∈M×I×Iandr∈R.

Proof. We showρ_i(xr,a,b)=ρ_i(x,ra,b). The proof thatρ_i(x,ar,b)=ρ_i(x,a,rb) is similar and so is omitted. Expandingⁱ_j=0d_i−j(xr)⊗[_k^j₌₀δ_j−k(a)⊗δ_k(b)] by the first summa- tion, we have

di(xr)⊗ ₀

k=0

δ0₋k(a)⊗δk(b)

+di−1(xr)⊗ ₁

k=0

δ1₋k(a)⊗δk(b)

+di−2(xr)⊗ ₂

k=0

δ2−k(a)⊗δk(b)

+···+xr⊗ _i

k=0

δi−k(a)⊗δk(b)

.

(5.3)

Using (5.3) and the definition ofdi−jforj=0, 1,...,n, we get _i

s=0

di−s(x)δs(r)

⊗ ₀

k=0

δ0₋k(a)⊗δk(b)

+ _i−1

s=0

di−1₋s(x)δs(r)

⊗ ₁

k=0

δ1₋k(a)⊗δk(b)

+ _i−2

s=0

di−2−s(x)δs(r)

⊗ ₂

k=0

δ2−k(a)⊗δk(b)

+···+xr⊗ _i

k=0

δ_i−k(a)⊗δ_k(b)

(5.4)

which, by shifting subscripts, can be written as _i

s=0

di−s(x)δs(r)

⊗ ₀

k=0

δ0₋k(a)⊗δk(b)

+ _i

s=1

di−s(x)δs−1(r)

⊗ ₁

k=0

δ1₋k(a)⊗δk(b)

+ _i

s=2

d_i−s(x)δ_s−2(r)

⊗ ₂

k=0

δ2−k(a)⊗δ_k(b)

+···+xr⊗ _i

k=0

δ_i−k(a)⊗δ_k(b)

.

(5.5)

Using properties of tensor products, (5.5) becomes _i

s=0

di−s(x)

⊗ ₀

k=0

δs(r)δ0₋k(a)⊗δk(b)

+ _i

s=1

di−s(x)

⊗ ₁

k=0

δs−1(r)δ1₋k(a)⊗δk(b)

+ _i

s=2

di−s(x)

⊗ ₂

k=0

δs−2(r)δ2−k(a)⊗δk(b)

+···+x⊗ _i

k=0

rδ_i−k(a)⊗δ_k(b)

.

(5.6)

We now use (5.6) to compute the (i−t)th term, where 0≤t≤i. Each summand [ⁱ_s=udi−s(x)]⊗[^u_k=0δs−u(r)δu−k(a)⊗δk(b)] in (5.6) contains an (i−t)th term until t > u. These terms are

di−t(x)⊗ ₀

k=0

δt(r)δ0₋k(a)⊗δk(b)

+di−t(x)⊗ ₁

k=0

δt−1(r)δ1₋k(a)⊗δk(b)

+di−t(x)⊗ ₂

k=0

δt−2(r)δ2₋k(a)⊗δk(b)

+···+d_i−t(x)⊗ _t

k=0

rδ_t−k(a)⊗δ_k(b)

.

(5.7)