Transfinite normal and composition series of modules
aa r X i v : . [ m a t h . R T ] S e p TRANSFINITE NORMAL ANDCOMPOSITION SERIES OF MODULES.
R. A. Sharipov
Abstract.
Normal and composition series of modules enumerated by ordinal num-bers are studied. The Jordan-H¨older theorem for them is discussed.
1. Introduction and some preliminaries.
Transfinite normal and composition series of groups were studied in [1]. Theygeneralize classical normal and composition series which are finitely long. In thispaper we reproduce the results of [1] in the case of left modules over associativerings and associative algebras.
Definition 1.1.
Let A be an associative ring and let V be an additive Abeliangroup. The Abelian group V is called a left module over the ring A (or a left A -module) if some homomorphism of rings ϕ : A → End( V )is given and fixed. Here End( V ) is the ring of endomorphisms of the additiveAbelian group V . Definition 1.2.
Let A be an associative algebra over some field K and let V be alinear vector space over the same field K . The linear vector space V is called a leftmodule over the algebra A (left A -module) if some homomorphism of algebras ϕ : A → End( V )is given and fixed. Here End( V ) is the algebra of endomorphisms of the linearvector space V .In both cases each element a ∈ A produces an operator ϕ ( a ) that can act uponany element v ∈ V . The result of applying ϕ ( a ) to v is denoted as u = ϕ ( a )( v ) . (1.1)In some cases the formula (1.1) is written as follows: u = ϕ ( a ) v. (1.2) Mathematics Subject Classification . 16D70. Typeset by
AMS -TEX R. A. SHARIPOV
The endomorphism ϕ ( a ) ∈ End( V ) in this formula is presented as a left multiplierfor the element v ∈ V . In some cases the symbol ϕ is also omitted: u = a v. (1.3)The formula (1.3) is a purely algebraic version of the formulas (1.1) and (1.2). Theaction of ϕ ( a ) upon v here is written as a multiplication of v on the left by theelement a ∈ A . This is the reason for which V is called a left A -module. Using(1.3), the definitions 1.1 and 1.2 can reformulated as follows. Definition 1.3.
A left module over an associative ring A is an additive Abeliangroup V equipped with an auxiliary operation of left multiplication by elements ofthe ring A such that the following identities are fulfilled:1) a ( v + v ) = a v + a v for all a ∈ A and for all v , v ∈ V ;2) ( a + a ) v = a v + a v for all a , a ∈ A and for all v ∈ V ;3) ( a a ) v = a ( a v ) for all a , a ∈ A and for all v ∈ V . Definition 1.4.
A left module over an associative K -algebra A is a linear vectorspace V over the field K equipped with an auxiliary operation of left multiplicationby elements of the algebra A such that the following identities are fulfilled:1) a ( v + v ) = a v + a v for all a ∈ A and for all v , v ∈ V ;2) ( a + a ) v = a v + a v for all a , a ∈ A and for all v ∈ V ;3) ( a a ) v = a ( a v ) for all a , a ∈ A and for all v ∈ V ;4) ( k a ) v = a ( k v ) = k ( a v ) for all k ∈ K , a ∈ A , and v ∈ V .Right A -modules are similar to left ones. Here elements v ∈ V are multiplied byelements a ∈ A on the right. However, each right A -module can be treated as a left A • -module, where A • is the opposite ring (opposite algebra) for A (see Chapter 5in [2]). For this reason below we consider left modules only.
2. Submodules and factormodules.
Definition 2.1.
Let V be a left A -module, where A is a ring. A subset W ⊆ V iscalled a submodule of V if it is closed with respect to the inversion, with respectto the addition, and with respect to the multiplication by elements of the ring A :1) v ∈ W implies − v ∈ W ;2) v ∈ W and v ∈ W imply ( v + v ) ∈ W ;3) v ∈ W and a ∈ A imply ( a v ) ∈ W .In other words, a submodule W is a subgroup of the additive group V invariantwith respect to the endomorphisms ϕ ( a ) for all a ∈ A . Definition 2.2.
Let V be a left A -module, where A is some K -algebra. A subset W ⊆ V is called a submodule of V if it is closed with respect to the addition andwith respect to the multiplication by elements of the field K and the algebra A :1) v ∈ W and k ∈ K imply ( k v ) ∈ W ;2) v ∈ W and v ∈ W imply ( v + v ) ∈ W ;3) v ∈ W and a ∈ A imply ( a v ) ∈ W .In other words, a submodule W is a subspace of the vector space V invariant withrespect to the endomorphisms ϕ ( a ) for all a ∈ A . RANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES. 3
In both cases a submodule W ⊆ V inherits the structure of a left A -module from V . Each submodule W is associated with the corresponding factorset V /W . Thefactorset
V /W is composed by cosetsCl W ( v ) = { u ∈ V : u = v + w for some w ∈ W } . (2.1)The element v in (2.1) is a representative of the coset Cl W ( v ). Any element v ∈ Cl W ( v ) can be chosen as its representative. In both cases (where A is a ring orwhere A is an algebra) the factorset V /W inherits the structure of a left A -modulefrom V . For this reason it is called a factormodule. Algebraic operations withcosets are given by the formulas1) Cl W ( v ) + Cl W ( v ) = Cl W ( v + v ) for any v , v ∈ V ;2) a Cl W ( v ) = Cl W ( a v ) for any a ∈ A and for any v ∈ V .If A is a K -algebra, then3) k Cl W ( v ) = Cl W ( k v ) for any k ∈ K and for any v ∈ V .
3. Transfinite normal and composition series.
Definition 3.1.
Let V be a left A -module. A transfinite sequence of its submodules { } = V V . . . V n = V (3.1)is called a transfinite normal series for V if V α = [ β<α V β for any limit ordinal α n .The term “normal series” in the above definition comes from the group theorywhere each normal series { } = G G . . . G n = G of a group G should becomposed by its subgroups such that G i is a normal subgroup of G i +1 . Otherwisewe would not be able to build the factorgroup G i +1 /G i . In the case of moduleseach submodule V i with i < n produces the factormodule V i +1 /V i . So the term“normal series” here is vestigial, it is used for ear comfort only. Definition 3.2.
A module V is called hypertranssimple if it has no normal series(neither finite nor transfinite) other than trivial one { } = V V = V . Definition 3.3.
A transfinite normal series (3.1) of a module V is called a trans-finite composition series of V if for each ordinal number i < n the factormodule V i +1 /V i is hypertranssimple.The concept of hypertranssimplicity is nontrivial in the case of groups. In thecase of modules each nontrivial submodule W of V produces the nontrivial normalseries { } = V V V = V , where V = W . For this reason the concept ofhypertranssimplicity of modules reduces to the standard concept of simplicity. Thedefinitions 3.2 and 3.3 then are reformulated as follows. Definition 3.4.
A module V is called simple if it has no submodules other thantrivial ones V = { } and V = V . Definition 3.5.
A transfinite normal series (3.1) of a module V is called a trans-finite composition series of V if for each ordinal number i < n the correspondingfactormodule V i +1 /V i is simple. R. A. SHARIPOV
4. Intersections and sums of submodules.
The intersection of two or more submodules of a given module V is again asubmodule of V . As for unions, the union of submodules in general case is not asubmodule. Unions of submodules are used to define their sums. Definition 4.1.
Let U i with i ∈ I be submodules of some module V . The sub-module U generated by the union of submodules U i is called their sum: U = X i ∈ I U i = D [ i ∈ I U i E . (4.1)If the number of submodules in (4.1) is finite, one can use the following notations: U = U + . . . + U n = h U ∪ . . . ∪ U n i . (4.2)In both cases (4.1) and (4.2) each element u ∈ U is presented as a finite sum u = u i + . . . + u i s , where u i r ∈ U i r and i r ∈ I for all r = 1 , . . . , s. (4.3) Definition 4.2.
The sum of submodules (4.1) is called a direct sum if for eachelement u ∈ U its presentation (4.3) is unique. Lemma 4.1 (Zassenhaus).
Let ˜ U and ˜ W be submodules of some left A -moduleand let U and W be submodules of ˜ U and ˜ W respectively. Then ( U + ( ˜ U ∩ ˜ W )) / ( U + ( ˜ U ∩ W )) ∼ = ( W + ( ˜ W ∩ ˜ U )) / ( W + ( ˜ W ∩ U )) . (4.4)The lemma 4.1 is also known as the butterfly lemma. Typically the butterflylemma is formulated for groups (see § Proof.
Let’s denote M = U + ( ˜ U ∩ W ) and N = W + ( ˜ W ∩ U ). Elements of thefactormodule in the left hand side of the formula (4.4) are cosets of the formCl M ( u + a ), where u ∈ U and a ∈ ˜ U ∩ ˜ W .
Note that U ⊆ M . Therefore Cl M ( u + a ) = Cl M ( a ) which means that each elementof the factormodule ( U + ( ˜ U ∩ ˜ W )) /M is represented by some element a ∈ ˜ U ∩ ˜ W .Thus we have a surjective homomorphism of modules ϕ : ˜ U ∩ ˜ W −→ ( U + ( ˜ U ∩ ˜ W )) /M. (4.5)Repeating the above arguments for the factormodule in the right hand side of theformula (4.4), we get another surjective homomorphism of modules ψ : ˜ U ∩ ˜ W −→ ( W + ( ˜ W ∩ ˜ U )) /N. (4.6)The rest is to prove that Ker ϕ = Ker ψ . Assume that a ∈ Ker ϕ . In this case a ∈ ˜ U ∩ ˜ W and a ∈ M . The inclusion a ∈ M means that a = u + w , where u ∈ U and w ∈ ˜ U ∩ W . Since ˜ U ∩ W ⊆ ˜ U ∩ ˜ W , from u = a − w we derive u ∈ ˜ U ∩ ˜ W . On RANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES. 5 the other hand u ∈ U . Hence u ∈ U ∩ ( ˜ U ∩ ˜ W ), which means u ∈ U ∩ ˜ W . Thus wehave proved that each element a ∈ Ker ϕ is presented as a sum a = u + w , where u ∈ ˜ W ∩ U and w ∈ ˜ U ∩ W. (4.7)Conversely, it is easy to see that the presentation (4.7) leads to a ∈ ˜ W ∩ ˜ U and a ∈ M , i. e. a ∈ Ker ϕ . For this reason we have Ker ϕ = ( ˜ W ∩ U ) + ( ˜ U ∩ W ). Theequality Ker ψ = ( ˜ W ∩ U ) + ( ˜ U ∩ W ) is proved similarly. Now the formula (4.4) isimmediate from Ker ϕ = Ker ψ due to the surjectivity of the homomorphisms (4.5)and (4.6). The butterfly lemma 4.1 is proved. (cid:3)
5. The Jordan-H¨older theorem.
Definition 5.1.
A transfinite normal series { } = ˜ V ˜ V . . . ˜ V p = G iscalled a refinement for a transfinite normal series { } = V V . . . V n = G ifeach submodule V i coincides with some submodule ˜ V j . Definition 5.2.
Two transfinite normal series { } = V V . . . V n = V and { } = W W . . . W m = V of a module V are called isomorphic if there is aone-to-one mapping that associates each ordinal number i < n with some ordinalnumber j < m in such a way that V i +1 /V i ∼ = W j +1 /W j . Theorem 5.1.
Arbitrary two transfinite normal series of a left A -module V haveisomorphic refinements. Lemma 5.1. If { } = G . . . G n = G is a transfinite composition series of agroup G , then it has no refinements different from itself. Theorem 5.2 (Jordan-H¨older).
Any two transfinite composition series of a left A -module V are isomorphic. The theorem 5.2 is immediate from the theorem 5.1 and the lemma 5.1. Asfor the theorem 5.1 and the lemma 5.1, their proof is quite similar to the proofof the theorem 3.1 and the lemma 3.10 in [1]. The algebraic part of this proof isbased on the Zassenhaus butterfly lemma. Its version for modules is given above(see lemma 4.1). The other part of the proof deals with indexing sets and ordinalnumbers, not with algebraic structures. For this reason it does not differ in thecase of groups and in the case of modules.
6. External direct sums.
Sums and direct sums introduced in the definitions 4.1 and 4.2 are internal ones.They are formed by submodules of a given module. External direct sums are formedby separate modules which are not necessarily enclosed in a given module.
Definition 6.1.
Let V i be left A -modules enumerated by elements i ∈ I of someindexing set I . Finite formal sums of the form v = v i + . . . + v i s , where v i r ∈ V i r and i r ∈ I for all r = 1 , . . . , s, (6.1)constitute a left A -module V which is called the direct sum of the modules V i .Once the external direct sum V is constructed, we find that it comprises sub-modules U i isomorphic to the initial modules V i . Indeed, we can set s = 1 in (6.1). R. A. SHARIPOV
Formal sums (6.1) with exactly one summand v = v i , where v i ∈ V i , constitute asubmodule U i of V isomorphic to the module V i . For this reason the constructionsof internal and external direct sums are the same in essential.According to the well-known Zermelo theorem (see Appendix 2 in [4]), everyset I can be well ordered and then associated with some ordinal number n (seeProposition 3.8 in Appendix 3 of [4]). Therefore the external direct sum V in theabove definition 6.1 can be written as V = M α Theorem 6.1. If a module V is presented as a direct sum of its simple submodules V i , then these submodules are unique up to the isomorphism and some permutationof their order in the direct sum. There is a special case of the external direct sum (6.2). Assume that U is somesimple left A -module. Let’s replicate this module into multiple copies and denotethese copies through V α . Then V α ∼ = U . In this case the module V in (6.2) isdenoted through N U , where N = | n | is the cardinality of the ordinal number n .Such a notation is motivated by the following theorem. Theorem 6.2. Let U be a simple left A -module and let N U and M U be twoexternal direct sums of the form (6.2) built by the copies of the module U : N U = M α N U is isomorphic to M U if and only if N = M , i. e. if | n | = | m | . The theorem 6.2 is easily derived from the theorem 6.1. 7. Concluding remarks. The results of this paper are rather obvious and are known to the algebraistscommunity. However, they are dispersed in various books as preliminaries to morespecial theories. Treated as obvious, these results are usually not equipped withexplicit proofs and even with explicit statements. We gather them in this paper forreferential purposes. RANSFINITE NORMAL AND COMPOSITION SERIES OF MODULES. 7 References 1. Sharipov R. A., Transfinite normal and composition series of groups , e-print arXiv:0908.2257in Electronic Archive http://arXiv.org.2. Cameron P. J., Introduction to algebra , Oxford University Press, New York, 2008.3. Lang S., Algebra , Springer-Verlag, New York, Berlin, Heidelberg, 2002.4. Grillet P. A., Abstract Algebra , Springer Science + Business Media, New York, 2007. E-mail address : [email protected] [email protected] URL ::