Draft lesson.

Ideals, quotients and torsion triples for many-object modules

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Killing an ideal of morphisms produces a full module subcategory with two explicit adjoints: a quotient on one side and an annihilator on the other. The ideal is idempotent exactly when that subcategory is closed under extensions. In that case it lies between two torsion pairs, and every such pair of adjacent torsion pairs arises uniquely this way. The two adjoints can still fail to be exact, and the torsion sequences can fail to split.

Use Modules and tensor functors over small preadditive categories for the pointwise abelian category and element-labelled representable presentations. Trace coreflections and balanced nonabelian categories proves the general probe-generated coreflection; Ideal-generated modules and nonstrict quotients gives a different, polynomial-ideal trace example. Our multiplication submodule below is defined by ideal arrows, rather than by all maps from a fixed probe. A freely accessible reference is Parra–Saorín–Virili, Torsion pairs in categories of modules over a preadditive category, §§2.1 and 4.2. We prove all constructions and both classification directions here.

1. Ideals and the quotient category

Fix a universe and a small preadditive category \(\mathcal C\). Right modules are additive functors \(M\colon\mathcal C^{\mathrm{op}}\to\mathsf{Ab}\), with group-homomorphism natural transformations as morphisms. All sums, products and indexing sets are small in this universe. Write \(\mathsf M=\mathsf{Mod}\text{-}\mathcal C\).

A two-sided ideal \(I\) consists of subgroups \(I(c,d)\subseteq\mathcal C(c,d)\), closed under precomposition and postcomposition by arbitrary arrows of \(\mathcal C\). Its square is the ideal

The empty sum is allowed. Precomposition and postcomposition retain this form, so it is an ideal; bilinearity gives subgroup closure. Always \(I^2\subseteq I\). Call \(I\) idempotent when equality holds, object by object.

The category \(\mathcal C/I\) has the same objects, morphism groups \(\mathcal C(c,d)/I(c,d)\), and the induced composition. This is well defined: changing either factor changes a composite by a sum of arrows in \(I\). Associativity and identity equations pass to the quotient, as does bilinearity. The quotient functor \(q\colon\mathcal C\to\mathcal C/I\) is additive. An identity may become zero; any module then has zero group at that object, since its identity map is also its zero map.

2. Multiplication and annihilator submodules

Define the following subgroups at each object:

Both are right submodules. For \(v\colon b\to c\), a generating value \(M(u)m\) of \((MI)(c)\) maps to \(M(v)M(u)m=M(uv)m\), which belongs to \((MI)(b)\) by precomposition closure. If \(m\) is annihilated at \(c\), then for \(t\in I(a,b)\) one has \(M(t)M(v)m=M(vt)m=0\), by postcomposition closure. This proves stability of the annihilator. The actions retain additivity, identities and composition from \(M\).

Every module morphism \(f\colon M\to N\) restricts to both submodules: naturality gives \(f_cM(u)m=N(u)f_dm\) for multiplication, and \(N(u)f_cm=f_bM(u)m=0\) for annihilators. Restriction preserves composition, identities and addition. Thus both constructions are additive functors, with natural inclusions into the identity functor.

Multiplication associates in the required sense:

Indeed, expanding a finite sum of values \(M(v)n\), where \(n\) is a finite sum of values \(M(u)m\), gives values \(M(uv)m\) with \(u,v\in I\). Conversely each value of a finite sum of such composites expands into a sum of these iterated values. This proves both subgroup inclusions at every object. Also \(h_dI=I(-,d)\) for \(h_d=\mathcal C(-,d)\): ideal closure gives one inclusion, while \(u=1_du\) gives the other.

Put \(\mathsf T_I=\{M:MI=0\}\). Equivalently, every action of an ideal arrow is zero. Quotients and submodules of such modules stay in \(\mathsf T_I\), as do all small products and coproducts, by their pointwise actions. Both \(M/MI\) and \(\operatorname{Ann}_I M\) belong to \(\mathsf T_I\): the quotient kills every ideal-arrow image, and the annihilator definition kills those images on its own elements.

3. The full restriction and its two adjoints

Restriction \(R=q^*\colon\mathsf{Mod}\text{-}(\mathcal C/I)\to\mathsf M\) is fully faithful, with essential image \(\mathsf T_I\). For an annihilated module the action of \([u]\) is \(M(u)\), independent of the representative; all group and functor laws descend. These operations give inverse module constructions. A group-homomorphism family is natural for the quotient arrows exactly when it is natural for their representatives, giving inverse constructions on every morphism as well. Restriction is exact by pointwise exactness.

Regard the quotient and annihilator as quotient-category modules and define

Their action on morphisms is induced quotient or restricted map, respectively. We have the actual adjoint triple \(L\dashv R\dashv A\). For \(N\) a quotient-category module, the two bijections are

For the first, compose a map \(LM\to N\) with the quotient \(\pi_M\colon M\to RLM\). Conversely, \(f\colon M\to RN\) kills \(MI\), since \(f_cM(u)m=RN(u)f_dm=0\) when \(u\in I\). Define its factor by \(\bar f_c([m])=f_c(m)\). This is a well-defined group homomorphism, and naturality follows by evaluating at representatives of every quotient arrow. Composing and factoring return the original map: on \(M\) this is the displayed formula, and on \(LM\) it follows from componentwise surjectivity of \(\pi_M\).

For the second, a map \(g\colon RN\to M\) has image in the annihilator, because \(M(u)g_c(n)=g_bRN(u)n=0\) for \(u\in I(b,c)\). Corestrict every component to obtain \(\bar g\colon N\to AM\); its naturality is the same equality inside the subgroups. The reverse map composes with \(\iota_M\colon RAM\hookrightarrow M\). Corestriction and composition are inverse because these inclusions are injective. Both bijections are natural in both variables: postcomposition and precomposition commute with the indicated formulas, quotient factors, and restrictions.

The first unit is \(\pi_M\); the second counit is \(\iota_M\). Since \(RN\) is annihilated, \(LRN\cong N\) by \([n]\mapsto n\), and \(ARN\cong N\) by the identity on its groups. These are the other counit and unit. The triangle equations follow directly: quotienting an annihilated group changes no element, and including its full annihilator changes no element. All four maps are natural module maps.

Thus \(L\) is right exact and \(A\) is left exact. To see these statements at the actual constructions, a map from a cokernel of \(f\) into \(RN\) corresponds to a map killing \(f\); (3.2) identifies it with a map from the cokernel of \(Lf\). Dually, maps \(RN\to\ker f\) correspond under (3.2) to maps into the kernel of \(Af\). The zero objects are preserved. No full exactness is implied, even for idempotent ideals.

4. Extension closure is precisely idempotence

Theorem 4.1. The class \(\mathsf T_I\) is closed under extensions in \(\mathsf M\) if and only if \(I^2=I\).

Proof. Suppose \(I^2=I\) and \(0\to N'\to M\to N''\to0\) is exact with both outer terms annihilated. Ideal-arrow values in \(M\) lie in \(N'\), since their images in \(N''\) vanish. A second ideal action kills them, so \(MI^2=0\) by (2.2). Idempotence gives \(MI=0\).

Conversely, for every \(d\) the pointwise exact sequence

has annihilated outer terms. The first is annihilated because precomposing one ideal arrow with another gives an arrow in \(I^2\); the last is annihilated by its definition. Extension closure makes the middle term annihilated. Apply any \(u\in I(c,d)\) to the class \([1_d]\) at \(d\). Its image \([u]\) at \(c\) must vanish, so \(u\in I^2(c,d)\). Together with \(I^2\subseteq I\), this proves equality. The argument includes a zero class \([1_d]\) without any exception. \(\square\)

5. The two canonical torsion pairs

A torsion pair \((\mathsf U,\mathsf V)\) in an abelian category consists of two classes closed under isomorphism, with \(\operatorname{Hom}(U,V)=0\) for all \(U\in\mathsf U,V\in\mathsf V\), such that every object has an exact row \(0\to U_M\to M\to V_M\to0\) with terms in these classes. A TTF triple, or torsion–torsionfree triple, is \((\mathsf C,\mathsf T,\mathsf F)\) for which both adjacent pairs are torsion pairs.

Assume now \(I^2=I\), and put

Theorem 5.1. The triple \((\mathsf C_I,\mathsf T_I,\mathsf F_I)\) is a TTF triple. Its torsion rows are the natural exact sequences

Proof. Equation (2.2) makes \(MI\) ideal-generated, and its quotient is annihilated. Any map from an ideal-generated module into an annihilated one kills every generating value \(M(u)m\) by naturality, hence is zero. This proves the first torsion pair.

The annihilator is annihilated. If \([m]\) in \(M/\operatorname{Ann}_I M\) is killed by every ideal arrow \(u\colon b\to c\), then \(M(u)m\) lies in the annihilator at \(b\). Consequently \(M(v)M(u)m=M(uv)m=0\) for every \(v\in I(a,b)\). Every ideal arrow into \(c\) is a finite sum of such composites, so \(I^2=I\) implies \(m\in(\operatorname{Ann}_I M)(c)\). Thus the quotient has zero annihilator. Finally, the image of a map from an annihilated module lies in the target's annihilator by the same calculation as in (3.2). If that annihilator is zero, the map is zero. This proves the second pair. Naturality of both rows was established in §2. \(\square\)

6. Recovering every TTF triple from representables

We first record the torsion-pair facts used in the converse, with their proofs. Write \({}^{\perp}\mathsf V\) for objects having no nonzero maps to any object of \(\mathsf V\), and \(\mathsf U^{\perp}\) for objects receiving no nonzero maps from \(\mathsf U\). The decomposition row of a torsion pair gives \(\mathsf U={}^{\perp}\mathsf V\) and \(\mathsf V=\mathsf U^{\perp}\): if \(M\) is left orthogonal, its epic map to \(V_M\) is zero, so \(V_M=0\); the dual argument uses the monic map from \(U_M\). The reverse inclusions are the defining orthogonality.

It follows that \(\mathsf U\) is closed under quotients and coproducts: a map from a quotient lifts along its epic quotient map, and a map from a coproduct is determined on its summands. Likewise \(\mathsf V\) is closed under subobjects and products, using a monic inclusion or all product projections. Both classes are extension closed. For an extension of two \(\mathsf U\)-objects, a map to \(V\) kills the subobject and factors through the quotient, so is zero. For an extension of two \(\mathsf V\)-objects, a map from \(U\) has zero composite to the quotient, hence factors through the subobject and is zero.

The torsion subobject \(tM=U_M\subseteq M\) is uniquely determined: every map from a \(\mathsf U\)-object into \(M\) has zero composite to \(V_M\) and hence factors uniquely through \(tM\). In particular it contains every \(\mathsf U\)-subobject of \(M\). This also defines \(t(f)\) as the restriction of any map \(f\colon M\to N\), and proves its additivity and functor laws. No sum over a possibly large class is required. One has \(tM=M\) exactly for \(M\in\mathsf U\), and \(tM=0\) exactly for \(M\in\mathsf V\).

Now let \((\mathsf C,\mathsf T,\mathsf F)\) be any TTF triple in \(\mathsf M\), and let \(t\) be the radical for its first pair \((\mathsf C,\mathsf T)\). The middle class \(\mathsf T\) is closed under quotients and coproducts because it is the torsion class of the second pair; it is also closed under subobjects, products and extensions by the preceding facts.

The radical \(t\) preserves coproducts. Indeed, sum the torsion rows pointwise. This yields an exact row with first term \(\bigoplus_j tM_j\in\mathsf C\) and last term \(\bigoplus_j(M_j/tM_j)\in\mathsf T\). Its first subobject is the torsion subobject by uniqueness. It also preserves epimorphisms. If \(f\colon M\twoheadrightarrow N\), then \(K=f(tM)\) belongs to \(\mathsf C\) by quotient closure, whereas \(N/K\), a quotient of \(M/tM\), belongs to \(\mathsf T\). Hence \(K=tN\), by uniqueness again, and \(t(f)\) is epic. Neither statement claims that \(t\) preserves ambient kernels.

Define

Submodule stability gives closure under precomposition. For \(v\colon d\to e\), the module map \(h_d\to h_e\) is postcomposition by \(v\); naturality of the subfunctor \(t\) sends \(t(h_d)\) into \(t(h_e)\). This gives postcomposition closure. Thus \(I\) is a two-sided ideal.

For every \(M\), the prerequisite supplies the epic evaluation map

Apply coproduct preservation and epimorphism preservation of \(t\). The image of \(\bigoplus_{(d,m)}I(-,d)\) under (6.2) is exactly \(MI\), by (2.1), and it is exactly \(tM\), by epicity of \(t(\epsilon_M)\). Therefore

Apply this to \(M=t(h_d)=I(-,d)\), which belongs to \(\mathsf C\). It gives \(I(-,d)I=I(-,d)\). The left side is \(I^2(-,d)\), by expanding compositions or by (2.2) and \(h_dI=I(-,d)\). Thus \(I^2=I\). Equation (6.3) also identifies \(\mathsf C=\mathsf C_I\) and \(\mathsf T=\mathsf T_I\). The second torsion class is now \(\mathsf T_I\), and its maximal torsion subobject is \(\operatorname{Ann}_I M\): it is annihilated, and every map from an annihilated object has image in it by §3. Hence its torsionfree class is \(\mathsf F_I\).

Starting instead with an idempotent ideal, the radical of the first pair in (5.2) is \(MI\). On \(h_d\) it gives \(I(-,d)\), so (6.1) recovers the original ideal. Starting with a TTF triple, the preceding argument recovers all three original classes. This proves a bijection between two-sided idempotent ideals and TTF triples in \(\mathsf{Mod}\text{-}\mathcal C\), with both inverse assignments explicitly proved.

There is no nonempty-object assumption: for empty \(\mathcal C\) the unique ideal is empty and the module category is the zero category, with its unique torsion triple. For \(I=0\) the triple is \((0,\mathsf M,0)\); for the full ideal it is \((\mathsf M,0,\mathsf M)\). The general theorem says nothing about splitting every torsion row or exactness of both quotient adjoints.

7. Four graded exercises with complete solutions

Exercise 1 (grade 1). Take the one-object category of \(\mathbb Z\), ideal \(I=2\mathbb Z\), and module \(M=\mathbb Z/4\). Compute both adjoints and their canonical maps. Show explicitly why extension closure and the second torsion row's torsionfree-quotient claim require idempotence.

Solution. Multiplication gives \(MI=2M=\{0,2\}\), so \(LM=M/2M\cong\mathbb Z/2\), with canonical map reduction modulo 2. The annihilator is \(AM=\{m:2m=0\}=\{0,2\}\cong\mathbb Z/2\), but its canonical map sends the generator to2 in \(M\). Thus the two abstract groups are isomorphic while their quotient and inclusion maps have different directions and meanings.

Here \(I^2=4\mathbb Z\ne I\). The exact row \(0\to\mathbb Z/2\xrightarrow{1\mapsto2}\mathbb Z/4\to\mathbb Z/2\to0\) has outer terms killed by 2 but middle term not killed by 2. This violates extension closure of \(\mathsf T_I\). Moreover \(M/AM\cong\mathbb Z/2\) has its entire group as annihilator; it is nonzero and is not in \(\mathsf F_I\). The quotient and annihilator adjunctions themselves remain valid.

Exercise 2 (grade 2). Let \(\mathcal C\) be the free preadditive path category \(0\xrightarrow{u}1\) over \(\mathbb Z\). A right module is \(M_1\xrightarrow{f}M_0\). For the ideal \(I_0\) generated by \(1_0\), compute \(MI_0,LM,AM\) and all three torsion classes. Give an explicit epimorphism not preserved by \(A\) and a nonsplit first torsion row.

Solution. The nonzero ideal groups are \(I_0(0,0)=\mathbb Z1_0\) and \(I_0(0,1)=\mathbb Zu\); \(I_0(1,1)=0\) and there are no arrows from 1 to 0. Both \(1_0=1_01_0\) and \(u=u1_0\) are products of ideal arrows, proving \(I_0^2=I_0\). Quotient modules have zero group at 0, so restriction sends \(N\) to \((N\to0)\).

The formulas (2.1) give \(MI_0=(0\to M_0)\), because the identity at 0 already generates every value there. Hence \(LM=M_1\). At0 the identity ideal action forces the annihilator to be zero; at 1 the ideal arrow \(u\) imposes \(f(m)=0\). Thus \(AM=\ker f\), embedded as \((\ker f\to0)\). The classes are \(\mathsf C_{I_0}=\{M_1=0\}\), \(\mathsf T_{I_0}=\{M_0=0\}\), and \(\mathsf F_{I_0}=\{f\text{ injective}\}\).

The map \(Q=(\mathbb Z\xrightarrow{1}\mathbb Z)\twoheadrightarrow N=(\mathbb Z\to0)\), with components identity at 1 and zero at 0, is natural and componentwise surjective. Its annihilator map is \(0\to\mathbb Z\), which is not surjective. Thus the right adjoint is not right exact, though the ideal is idempotent. The first torsion row of \(Q\) is \(0\to(0\to\mathbb Z)\to Q\to(\mathbb Z\to0)\to0\). A section would have component identity at 1 and zero at 0. Naturality along \(u\) would require the identity of \(\mathbb Z\) to equal zero, so no section exists.

Exercise 3 (grade 3). In the same path category use the ideal \(I_1\) generated by \(1_1\). Compute both adjoints and the triple. Show that the left adjoint need not preserve monomorphisms and that the class \(\mathsf C_{I_1}\) need not be closed under ambient kernels.

Solution. Now \(I_1(1,1)=\mathbb Z1_1\), \(I_1(0,1)=\mathbb Zu\), and the other ideal groups are zero. The factorizations \(1_1=1_11_1\) and \(u=1_1u\) show idempotence. Quotient restriction sends \(N\) to \((0\to N)\). One has \(MI_1=(M_1\xrightarrow{f}\operatorname{im}f)\), so \(LM=\operatorname{coker}f\); the annihilator is \((0\to M_0)\), so \(AM=M_0\). Thus \(\mathsf C_{I_1}=\{f\text{ surjective}\}\), \(\mathsf T_{I_1}=\{M_1=0\}\), and \(\mathsf F_{I_1}=\{M_0=0\}\).

The natural map \(P=(0\to\mathbb Z)\hookrightarrow Q=(\mathbb Z\xrightarrow{1}\mathbb Z)\) has zero component at 1 and identity at 0; it is componentwise injective. Its image under \(L\) is \(\mathbb Z\to0\), which is not injective. Hence idempotence does not make the left adjoint left exact.

The epimorphism \(Q\to N=(\mathbb Z\to0)\) from Exercise 2 has both source and target in \(\mathsf C_{I_1}\), since both displayed arrows are surjective. Its ambient kernel is \(P\), whose arrow \(0\to\mathbb Z\) is not surjective. Thus this torsion class is not closed under ambient kernels. In fact \(PI_1=0\), so its intrinsic coreflected kernel is zero. The coreflection factorization is verified directly: a map from an object \((B_1\twoheadrightarrow B_0)\) to \(M\) has image at 0 contained in \(\operatorname{im}f\), because its component at 0 is determined by its values along the surjection \(B_1\to B_0\). It therefore factors uniquely through \(MI_1\). This illustrates exactly why that right adjoint does not retain ambient kernels under inclusion.

Exercise 4 (grade 4). For the one-object ring \(R=k\times k\), let \(e=(1,0)\) and \(I=Re\). Prove that both torsion rows split naturally, compute the entire triple, and recover the ideal from the first radical on the representable. Explain why the quotient module category alone does not specify a torsion triple inside a fixed module category, using Exercises 2–3.

Solution. Here \(e\) is central and \(e^2=e\). Every right module has a natural direct decomposition \(M=Me\oplus M(1-e)\): their sum is all of \(M\) because \(e+(1-e)=1\), and their intersection is zero because multiplying an element in it by \(e\) gives both itself and zero. Multiplication by either idempotent is an \(R\)-linear projection, by centrality, and every module map commutes with these projections. The decomposition is consequently natural, including on all module morphisms.

We have \(MI=Me\), \(AM=M(1-e)\), and \(LM=M/Me\cong M(1-e)\) by \([m]\mapsto m(1-e)\). Its inverse includes \(M(1-e)\) and then takes the quotient. Both inverse identities follow from the decomposition. The first torsion row splits by this inclusion, and the second splits by including \(Me\) as representatives of \(M/M(1-e)\). Thus \(\mathsf C_I\) consists of modules supported in the first factor, \(\mathsf T_I\) of modules supported in the second, and \(\mathsf F_I\) again of modules supported in the first. All these identifications are natural, not just abstract group decompositions.

For the representable \(h=R_R\), the first torsion radical is \(t(h)=Re\), so evaluating (6.1) recovers the original ideal as the indicated subgroup of \(R\). This keeps its position inside the ambient ring. In the path category, the two quotient module categories of Exercises 2–3 are both equivalent to \(\mathsf{Ab}\), but their fully faithful restrictions are \(N\mapsto(N\to0)\) and \(N\mapsto(0\to N)\). These are distinct subcategories of the fixed path-module category. Their ideals and three torsion classes are distinct. The classification retains the embedded middle class, or equivalently all ideal subgroups; the abstract equivalence type of the quotient category alone loses that information. The nonsplit row in Exercise 2 also shows that idempotence without the central decomposition used here does not supply splitting.

Reading and attribution

Parra–Saorín–Virili, Torsion pairs in categories of modules over a preadditive category, version 2, 19 November 2019, §2.1 gives ideals, quotient categories and multiplication modules; Lemmas 4.3–4.4 and Theorem 4.5 give the idempotent-ideal/TTF correspondence. Original lesson text: CC0-1.0.