aa r X i v : . [ m a t h . R A ] J a n Noncommutative Henselizations
Masood Aryapoor
Division of Mathematics and PhysicsM¨alardalen UniversityHamngatan 15, 632 17, Eskilstuna, Sweden
Abstract
In this paper, the familiar notion of a Henselian pair is extended tothe noncommutative case. Furthermore, the problem of Henselizationsis studied in the noncommutative context, and it is shown that every(not necessarily commutative) pair which is Hausdorff with respect toa certain topology has a left (and right) Henselization.
Keywords:
Noncommutative Henselian pair, Noncommutative Henseliza-tion
The notion of a Henselian ring, introduced by Azumaya in [3], is well-knownin the commutative case. This concept has been extended to the noncommu-tative case, see [1]. However, the theory of noncommutative Henselian ringsis not as well-studied as its commutative counterpart, and many problemsconcerning noncommutative Henselian rings are still open. One such prob-lem, discussed in [1], is the problem of noncommutative Henselizations. Inthe commutative case, it is known that every local ring has a Henselization,see [8]. The aim of this article is to investigate the notion of Henselizationin the noncommutative context. One of our results is that every noncom-mutative local ring satisfying a kind of “commutativity” condition has aHenselization, see Subsection 3.7.In Section 2, we present the preliminaries. In particular, the notion of anoncommutative Henselian ring, introduced in [1], is generalized in two dif-ferent ways. First, we introduce the notions of left Henselian rings and rightHenselian rings, which is more natural in the noncommutative context. Sec-ond, this concept is generalized to pairs as done by Lafon in the commutativecase, see [6]. The final section is devoted to the notions of left Henselizationsand right Henselizations. It is shown that every perfect pair has a left (andright) Henselization which is unique up to unique isomorphism.1
Henselian pairs
In this section, we present the preliminaries, and in particular, the notions ofleft Heneslian rings and right Henselian rings. In this paper, we shall assumethat all rings have a unit, and all ring homomorphisms are unit-preserving.
The category of pairs has been introduced in the commutative setting, see[6]. One can enlarge this category to include noncommutative rings. Moreprecisely, the objects of the category P of pairs are pairs ( A, I ) where A isa (unitary but not necessarily commutative) ring and I is an ideal of A . Amorphism φ : ( A, I ) → ( B, J ) of pairs is a ring homomorphism φ : A → B satisfying φ − ( J ) = I . It is straightforward to verify that any directedsystem in P has a direct limit. Likewise, any inverse system in P has aninverse limit.For a pair ( A, I ), we denote the image of a ∈ A in A/I by ¯ a . Givena morphism φ : ( A, I ) → ( B, J ) of pairs, we have a canonical homomor-phism ¯ φ : A/I → B/J induced by φ . The proof of the following lemma isstraightforward and left to the reader. Lemma 2.1.
Let φ : ( A, I ) → ( C, K ) and ψ : ( B, J ) → ( C, K ) be mor-phisms of pairs. If there is a ring homomorphism α : A → B such that φ = ψ ◦ α as ring homomorphisms, then α : ( A, I ) → ( B, J ) defines amorphism of pairs. For a ring A , the ring of polynomials F ( x ) = P ni =0 a i x i over A , where theindeterminate x commutes with all a ∈ A , is denoted by A [ x ]. For a subset S ⊂ A , the set of all polynomials P ni =0 a i x i ∈ A [ x ], where a , ..., a n ∈ S , isdenoted by S [ x ].Two polynomials F ( x ) , F ( x ) ∈ A [ x ] are called left coprime if A [ x ] F ( x ) + A [ x ] F ( x ) = A [ x ] . The notion of right coprime polynomials is defined in the obvious way. Itis easy to see that polynomials F ( x ) , F ( x ) ∈ A [ x ] are left coprime if andonly if there are polynomials G ( x ) , G ( x ) ∈ A [ x ] such that G ( x ) F ( x ) + G ( x ) F ( x ) = 1 . For a pair (
A, I ), we denote the canonical homomorphism A [ x ] → ( A/I )[ x ]by F ( x ) = P i a i x i ¯ F ( x ) = P i ¯ a i x i .2 .3 Euclidean algorithm Let A be a ring. The following version of Euclidean algorithm holds in A [ x ]. Lemma 2.2.
Let I be a right ideal of A . For every monic polynomial F ( x ) ∈ A [ x ] and every polynomial G ( x ) ∈ I [ x ] , there exist polynomials Q ( x ) , R ( x ) ∈ I [ x ] such that G ( x ) = Q ( x ) F ( x ) + R ( x ) and deg( R ( x )) < deg( F ( x )) . Proof.
We use induction on deg( G ( x )). If deg( G ( x )) < deg( F ( x )), we set Q ( x ) = 0 and R ( x ) = G ( x ), so we are done. Let G ( x ) = P mi =0 b i x i and F ( x ) = a + a x + · · · + a n − x n − + x n , where m ≥ n . The polynomial G ( x ) = G ( x ) − b m x m − n F ( x ) belongs to I because b m ∈ I and I is a right ideal. Since deg( G ( x )) < deg( G ( x )) , byinduction, there are polynomials Q ( x ) , R ( x ) ∈ I [ x ] such that G ( x ) = Q ( x ) F ( x ) + R ( x ) and deg( R ( x )) < deg( F ( x )) . Setting Q ( x ) = b m x m − n + Q ( x ), we obtain G ( x ) = Q ( x ) F ( x ) + R ( x ) , where Q ( x ) , R ( x ) satisfy the desired conditions, and we are done. A pair (
A, I ) is called a
Jacobson pair if I ⊂ rad ( A ) where rad ( A ) is theJacobson radical of A . We have the following result concerning Jacobsonpairs. Lemma 2.3.
Let ( A, I ) be a Jacobson pair. Then, monic polynomials F ( x ) , F ( x ) ∈ A [ x ] are left (resp. right) coprime if and only if the poly-nomials ¯ F ( x ) , ¯ F ( x ) ∈ ( A/I )[ x ] are left (resp. right) coprime.Proof. The “only if” part is trivial. To prove the other direction, supposethat ¯ F ( x ) , ¯ F ( x ) ∈ ( A/I )[ x ] are left coprime. It follows that A [ x ] F ( x ) + A [ x ] F ( x ) + I [ x ] = A [ x ]Considering M = A [ x ] A [ x ] F ( x )as a left A -module in the obvious way, we see that M = N + IM where N is the submodule of M generated by F ( x ). The module M is a finitelygenerated A -module because F ( x ) is monic. Since I ⊂ rad ( A ), Nakayama’slemma implies that N = M , that is, A [ x ] F ( x ) + A [ x ] F ( x ) = A [ x ] , and we are done. 3 .5 Local homomorphisms A ring homomorphism φ : A → B is called local if φ sends every nonunit in A to a nonunit in B . Here, we provide some facts concerning local maps.The proof of the first lemma is easy and left to the reader. Lemma 2.4.
Let φ : A → B and ψ : B → C be ring homomorphisms. If ψ ◦ φ is local, then φ is local too. Lemma 2.5.
Let φ : A → B be a local homomorphism. If B is a local ring,then A is a local ring whose maximal ideal is φ − ( I ) where I is the maximalideal of B .Proof. To show that A is a local ring, we need to prove that if a + b isinvertible in A for some a, b ∈ A , then either a or b is invertible in A , seeTheorem 19.1 in [7]. If a + b is invertible in A for some a, b ∈ A , then φ ( a ) + φ ( b ) is invertible in B which implies that φ ( a ) or φ ( b ) is invertible in B , because B is local. It follows that either a or b is invertible in A because φ is local. Therefore, A is a local ring. Since φ is a local homomorphismand B is a local ring, we see that φ − ( I ) consists of all nonunits in A , thatis, φ − ( I ) is the maximal ideal of A . Let φ : A → B be a ring homomorphism. In what follows, we introduce aring A φ , a ring homomorphism Λ φ : A → A φ and a local homomorphismΨ φ : A φ → B such that φ = Ψ φ ◦ Λ φ . Let S be the set of all elements s ∈ A such that ψ ( s ) is invertible in B . Let A = A S be the localizationof A at S and λ : A → A be the canonical homomorphism, consult [4]for the definition and elementary properties of localizations. It follows fromthe universal property of A S that there exists a unique homomorphism ψ : A → B such that φ = ψ ◦ λ . Let S be the set of all elements s ∈ A such that φ ( s ) is invertible in B . Consider the localization A = ( A ) S of A at S and let λ : A → A be the canonical homomorphism. Itfollows from the universal property of A that there exists a homomorphism ψ : A → B such that ψ = ψ ◦ λ . Continuing this process, we obtaina sequence of rings A , A , A , ... , homomorphisms λ i : A i − → A i and ψ i : A i → B such that ψ i − = ψ i ◦ λ i for i = 1 , , ..., where ψ = φ . Weset A φ = lim −→ A i . We have a canonical homomorphism Λ φ : A → A φ and acanonical homomorphism Ψ φ : A φ → B . In the following proposition, weprovide some facts about this construction, see Section 2 in [2] for a proofof this proposition and a detailed discussion of the localization ring A φ . Proposition 2.6.
The ring homomorphism Ψ φ : A φ → B is local andsatisfies the equality φ = Ψ φ ◦ Λ φ . .7 Commutativity with respect to a filtration Let (
A, I ) be a pair. A descending sequence F of ideals I ⊃ I ⊃ ... of A is called a filtration on ( A, I ) if I = I . The pair ( A, I ) is called commutative with respect to F , or F - commutative for short, if [ A, I n ] ⊂ I n +1 for all n = 1 , , ... . The notation [ S, T ], where
S, T ⊂ A , stands for the setof all elements of the form st − ts where s ∈ S, t ∈ T . For every pair (
A, I ), one can define a sequence I (1) , I (2) , ... of ideals of A asfollows: I (1) = I ; for n = 1 , , ... , the ideal I ( n +1) is the ideal generated by[ A, I ( n ) ]. It is easy to see that the sequence I (1) , I (2) , ... is a filtration on ( A, I )called the commutator filtration of (
A, I ). Clearly, (
A, I ) is commutativewith respect to its commutator filtration. Note that if φ : ( A, I ) → ( B, J )is a morphism of pairs then φ ( I ( n ) ) ⊂ J ( n ) for all n = 1 , , ... . Let (
A, I ) be a pair and F : I = I, I , ... be a filtration on ( A, I ). The F - topology on ( A, I ) is the linear topology on A for which the sets I, I , . . . form a fundamental system of neighborhoods of 0. The pair ( A, I ) is called separated with respect to F , or F -separated for short, if A is Hausdorff withrespect to the F -topology. Note that ( A, I ) is F -separated if and only if ∩ ∞ n =1 I n = { } . The pair ( A, I ) is called complete with respect to F , or F -complete for short, if it is complete with respect to the F -topology. A pair (
A, I ) is called a left unique factorization pair , or
LUFP for short, iffor every factorization ¯ F ( x ) = f ( x ) f ( x ) of a monic polynomial F ( x ) ∈ A [ x ]over A/I , where f ( x ) f ( x ) ∈ ( A/I )[ x ] are left coprime monic polynomi-als, there exists at most one factorization F ( x ) = F ( x ) F ( x ) such that F ( x ) , F ( x ) ∈ A [ x ] are monic polynomials, and ¯ F ( x ) = f ( x ) , ¯ F ( x ) = f ( x ). The notion of a right unique factorization pair (RUFP) is defined ina similar way. A pair is called a unique factorization pair , or UFP for short,if it is both an LUFP and an RUFP.
A pair (
A, I ) is called left Henselian if (
A, I ) is a Jacobson pair, and thefollowing version of Hensel’s lemma holds in A. For every monic polyno-mial F ( x ) ∈ A [ x ], if ¯ F ( x ) = f ( x ) f ( x ), where f ( x ) , f ( x ) ∈ ( A/I )[ x ] are5eft coprime monic polynomials, then there exist unique monic polynomi-als F ( x ) , F ( x ) ∈ A [ x ] satisfying F ( x ) = F ( x ) F ( x ), ¯ F ( x ) = f ( x ) and¯ F ( x ) = f ( x ). We note that the polynomials F ( x ) and F ( x ) are left co-prime, see Lemma 2.3. The notion of a right Henselian pair is defined ina similar fashion. A pair which is both left and right Henselian is called Henselian . Obviously, every (resp. left or right) Henselian ring is a (resp.left or right) UFP. We note that if
A/I is a commutative ring, then (
A, I )is left Henselian if and only if it is right Henselian.
The following result generalizes Theorem 2.1 in [1].
Theorem 2.7.
Let ( A, I ) be a pair and F : I = I, I , ... be a filtrationon ( A, I ) such that ( A, I ) is F -commutative, F -separated and F -complete.If I n [ A, A ] ⊂ I n +1 and I n ⊂ I n +1 for all n = 1 , , ... , then ( A, I ) is leftHenselian.Proof. First, we show that (
A, I ) is a Jacobson pair. Let a ∈ I be given.The condition I n ⊂ I n +1 for all n = 1 , , ... , implies that the geometric series1 − a + a − · · · converges to a unique limit in A because ( A, I ) is F -separated and F -complete. The limit of this series is the inverse of 1 + a . Therefore, everyelement in 1 + I is invertible, which implies that I ⊂ rad ( A ), that is, ( A, I )is a Jacobson pair.To show that (
A, I ) is left Henselian, let F ( x ) ∈ A [ x ] be a monic poly-nomial such that ¯ F ( x ) = f ( x ) f ( x ) where f ( x ) , f ( x ) ∈ ( A/I )[ x ] are leftcoprime monic polynomials. We need to show that we can lift this factor-ization to A [ x ] in a unique way. First, we show that there are sequences ofmonic polynomials F , ( x ) , F , ( x ) , ..., F ,i ( x ) , ... and F , ( x ) , F , ( x ) , ..., F ,i ( x ) , ... in A [ x ] such that ¯ F ,i ( x ) = f ( x ) , ¯ F ,i ( x ) = f ( x ) ,F ,i +1 ( x ) − F ,i ( x ) ∈ I i [ x ] , F ,i +1 ( x ) − F ,i ( x ) ∈ I i [ x ] ,F ( x ) − F ,i ( x ) F ,i ( x ) ∈ I i [ x ] , for all i ≥
1. To construct these sequences, we use induction on i . Since thecanonical map A → A/I is onto, we can find monic polynomials F , ( x ) , F , ( x ) ∈ A [ x ]6uch that ¯ F , ( x ) = f ( x ) , ¯ F , ( x ) = f ( x ). Clearly, F , ( x ) and F , ( x )satisfy the desired conditions. Having found F ,i ( x ) and F ,i ( x ), we find F ,i +1 ( x ) and F ,i +1 ( x ) as follows. Set G ( x ) = F ( x ) − F ,i ( x ) F ,i ( x ) . I claim that there are polynomials R ( x ) , R ( x ) ∈ I i [ x ] such thatdeg( R ( x )) < deg( F ,i ( x )) , deg( R ( x )) < deg( F ,i ( x )) , and R ( x ) F ,i ( x ) + R ( x ) F ,i ( x ) − G ( x ) ∈ I i +1 [ x ]By Lemma 2.3, there exit polynomials H ( x ) , H ( x ) ∈ A [ x ] such H ( x ) F ,i ( x ) + H ( x ) F ,i ( x ) = 1 . It follows that G ( x ) F ,i ( x ) + G ( x ) F ,i ( x ) = G ( x ) , where both G ( x ) = G ( x ) H ( x ) and G ( x ) = G ( x ) H ( x ) belong to I i [ x ].By Lemma 2.2, there are polynomials Q ( x ) , R ( x ) ∈ I i [ x ] such that G ( x ) = Q ( x ) F ,i ( x ) + R ( x ) and deg( R ( x )) < deg( F ,i ( x )) . It follows that G ( x ) = R ( x ) F ,i ( x ) + ( G ( x ) + Q ( x ) F ,i ( x )) F ,i ( x )+ Q ( x )( F ,i ( x ) F ,i ( x ) − F ,i ( x ) F ,i ( x ))Using the condition I i [ A, A ] ⊂ I i +1 , we see that R ( x ) F ,i ( x ) + ( G ( x ) + Q ( x ) F ,i ( x )) F ,i ( x ) − G ( x ) ∈ I i +1 [ x ] . Let G ( x ) + Q ( x ) F ,i ( x ) = P mi =0 b i x i . Assume that m ≥ deg( F ,i ( x )). Since F ,i ( x ) is monic, anddeg( R ( x ) F ,i ( x )) < deg( F ,i ( x )) + deg( F ,i ( x )) , deg( G ( x )) < deg( F ,i ( x )) + deg( F ,i ( x )) , the relation R ( x ) F ,i ( x ) + ( G ( x ) + Q ( x ) F ,i ( x )) F ,i ( x ) − G ( x ) ∈ I i +1 [ x ]implies that b m ∈ I i +1 . Therefore, we have R ( x ) F ,i ( x ) + ( G ( x ) − b m x m ) F ,i ( x ) − G ( x ) ∈ I i +1 [ x ] . R ( x ) F ,i ( x ) + R ( x ) F ,i ( x ) − G ( x ) ∈ I i +1 [ x ] , where R ( x ) = P i< deg( F ( x )) b i x i , proving the claim. We set F ,i +1 ( x ) = F ,i ( x ) + R ( x ) ,F ,i +1 ( x ) = F ,i ( x ) + R ( x ) . Clearly, we have ¯ F ,i +1 ( x ) = f ( x ) , ¯ F ,i +1 ( x ) = f ( x ) F ,i +1 ( x ) − F ,i ( x ) ∈ I i [ x ] , F ,i +1 ( x ) − F ,i ( x ) ∈ I i [ x ]Moreover, we have F ( x ) − F ,i +1 ( x ) F ,i +1 ( x ) =( F ( x ) − F ,i ( x ) F ,i ( x )) − F ,i ( x ) R ( x ) − R ( x ) F ,i ( x ) − R ( x ) R ( x ) =( G ( x ) − R ( x ) F ,i ( x ) − R ( x ) F ,i ( x ))+( R ( x ) F ,i ( x ) − F ,i ( x ) R ( x )) − R ( x ) R ( x ) . Since (
A, I ) is F -commutative and I i ⊂ I i +1 , we deduce that F ( x ) − F ,i +1 ( x ) F ,i +1 ( x ) ∈ I i +1 [ x ] . Having constructed the desired sequences, we proceed as follows. Since A is F -complete, the limits F ( x ) = lim i →∞ F ,i ( x ) and F ( x ) = lim i →∞ F ,i ( x )exist. Clearly, we have ¯ F ( x ) = f ( x ) and ¯ F ( x ) = f ( x ). Since A is F -Hausdorff, we have F ( x ) = F ( x ) F ( x ) and F , F are monic polynomials.Since ( A, I ) is F -commutative, it is easy to see that I ( n ) ⊂ I n for all n . Itfollows that ( A, I ) is also separated with respect to its commutator filtration.Therefore, by Proposition 3.1, this factorization is unique, and we are done.
The notion of the Henselization of a pair has been introduced in the com-mutative case, see [6]. It turns out that one can develop a similar theoryin the noncommutative case. However, we focus our attention on a specialsubcategory P of the category P of pairs and prove that every object inthis subcategory has a left (and a right) Henselization in P , see Theorem3.5. Our treatment of Henselization is somewhat similar to the one given in[5]. 8 .1 The category of perfect pairs A pair (
A, I ) is called perfect if (
A, I ) is separated with respect to its com-mutator filtration, that is, ∩ ∞ n =1 I ( n ) = { } . The full subcategory of thecategory P consisting of perfect pairs is denoted by P . For any pair ( A, I ),it is easy to see that the pair ̥ ( A, I ) = ( A ∩ ∞ n =1 I ( n ) , I ∩ ∞ n =1 I ( n ) )is a perfect pair. Furthermore, the assignment ( A, I ) ̥ ( A, I ) gives riseto a functor ̥ : P → P . One can easily check that ̥ is a left adjoint ofthe inclusion function ι : P → P . The reason for restricting our attention to P is in the following result. Proposition 3.1.
Every perfect Jacobson pair is a UFP.Proof.
Let (
A, I ) be a perfect Jacobson pair. We only show that (
A, I ) is anLUFP. The proof that A is an RUFP is similar. Assume, on the contrary,that there are different factorizations F ( x ) = F ( x ) F ( x ) = G ( x ) G ( x )of a monic polynomial F ( x ) ∈ A [ x ] such that f ( x ) = ¯ F ( x ) = ¯ G ( x ) and f ( x ) = ¯ F ( x ) = ¯ G ( x ) are left coprime monic polynomials. Without lossof generality, we may assume F ( x ) = G ( x ). The facts that F ( x ) and G ( x ) are monic polynomials, and ¯ F ( x ) = ¯ G ( x ), imply that there exists apolynomial N ( x ) ∈ I [ x ] such that F ( x ) = G ( x ) + N ( x ) and deg( N ( x )) < deg( G ( x )) . Since ∩ ∞ n =1 I ( n ) = { } and N ( x ) = 0, there exists d ≥ N ( x ) ∈ I ( d ) [ x ] but N ( x ) / ∈ I ( d +1) [ x ] . Since ¯ F ( x ) and ¯ G ( x ) are left coprime, it follows from Lemma 2.3 that thereare polynomials H ( x ) , H ( x ) ∈ A [ x ] such that H ( x ) F ( x ) + H ( x ) G ( x ) = 1 . We can write F ( x ) = H ( x ) F ( x ) F ( x ) + H ( x ) G ( x ) F ( x ) =( H ( x ) G ( x ) + H ( x ) F ( x )) G ( x ) + H ( x )( G ( x ) N ( x ) − N ( x ) G ( x )) . K ( x ) = H ( x ) G ( x ) + H ( x ) F ( x ), we see that F ( x ) = K ( x ) G ( x )as elements of ( A/I ( d +1) )[ x ], because ( A, I ) is commutative with respectto its commutator filtration. Since F ( x ) and G ( x ) are monic polynomi-als of the same degree, we conclude that F ( x ) = G ( x ) as elements of( A/I ( d +1) )[ x ], that is, N ( x ) = F ( x ) − G ( x ) ∈ I ( d +1) [ x ] , a contradiction . In this part, we prove the following result.
Proposition 3.2.
Let ( A, I ) be a perfect pair and F ( x ) ∈ A [ x ] be a monicpolynomial. Suppose that ¯ F ( x ) has a factorization ¯ F ( x ) = f ( x ) f ( x ) over A/I where f ( x ) , f ( x ) ∈ ( A/I )[ x ] are left coprime monic polynomials. Then,there exists a perfect Jacobson pair ( A h F ; f , f i , I h F ; f , f i ) and a mor-phism Φ h F ; f ,f i : ( A, I ) → ( A h F ; f , f i , I h F ; f , f i ) of pairs having the following universal property. For every morphism φ : ( A, I ) → ( B, K ) of pairs, where ( B, K ) is a perfect Jacobson pair, if φ ( F ( x )) = G ( x ) G ( x ) for some monic polynomials G ( x ) , G ( x ) ∈ B [ x ] such that ¯ G ( x ) = ¯ φ ( f ( x )) , ¯ G ( x ) = ¯ φ ( f ( x )) , then there exists a unique morphism ψ : ( A h F ; f , f i , I h F ; f , f i ) → ( B, K ) of pairs such that φ = ψ ◦ Φ h F ; f ,f i .Proof. First, we give a construction of the pair ( A h F ; f , f i , I h F ; f , f i )after which we prove its universal property. Let F ( x ) = a + a x + · · · + a d − x d − + x d ,f ( x ) = b + b x + · · · + b d − x d − + x d ,f ( x ) = c + c x + · · · + c d − x d − + x d . We consider the free A -ring A h y , ..., y d − , z , ..., z d − i generated by thenoncommutating variables y , ..., y d − , z , ..., z d − . We have( y + y x + · · · + y d − x d − + x d )( z + z x + · · · + z d − x d − + x d ) = g + g x + · · · + g d − x d − + x d g , ..., g d − ∈ A h y , ..., y d − , z , ..., z d − i . Consider the ring homo-morphism α : A h y , ..., y d − , z , ..., z d − i → AI defined by α ( y ) = b , ..., α ( y d − ) = b d − , α ( z ) = c , ..., α ( z d − ) = c d − ,α ( a ) = ¯ a where a ∈ A. The ideal h g − a , ..., g d − − a d − i generated by g − a , ..., g d − − a d − iscontained in the kernel of α because ¯ F ( x ) = f ( x ) f ( x ). Therefore, α givesrise to a ring homomorphism β : A h y , ..., y d − , z , ..., z d − ih g − a , ..., g d − − a d − i → AI Consider the following localization ring (see subsection 2.6) R = (cid:16) A h y , ..., y d − , z , ..., z d − ih g − a , ..., g d − − a d − i (cid:17) β . We have canonical ring homomorphisms γ : R → AI and η : A → R which satisfy γ ( η ( a )) = ¯ a for every a ∈ A , see Proposition 2.6. We set J = ker( γ ). By Proposition 2.6, γ is local from which it follows that J ⊂ rad ( A h F ; f , f i ), that is, ( R, J ) is a Jacobson pair. Since the quo-tient homomorphism A → A/I is a morphism (
A, I ) → ( A/I,
0) of pairs,we can use Lemma 2.1 to deduce that η : ( A, I ) → ( R, J ) is a morphism ofpairs. Finally, we set( A h F ; f , f i , I h F ; f , f i ) = ̥ ( R, J )The canonical morphism η : ( A, I ) → ( R, J ) composed with the quotientmorphism (
R, I ) → ̥ ( R, J ) gives a morphismΦ h F ; f ,f i : ( A, I ) → ( A h F ; f , f i , I h F ; f , f i )of pairs.To prove the universal property of Φ h F ; f ,f i , let φ : ( A, I ) → ( B, K ) bea morphism of pairs where (
B, K ) is a perfect Jacobson pair. Suppose that φ ( F )( x ) = G ( x ) G ( x ) where G ( x ) , G ( x ) ∈ B [ x ] are monic polynomialsand ¯ G = ¯ φ ( f ) , ¯ G = ¯ φ ( f ) . Let G ( x ) = e + e x + · · · + e d − x d − + x d ,G ( x ) = f + f x + · · · + f d − x d − + x d . y e , ..., y d − e d − , z f , ..., z d − f d − yield a ring homomorphism ψ : A h y , ..., y d − , z , ..., z d − ih g − a , ..., g d − − a d − i → B which extends the ring homomorphism φ . Furthermore, since K ⊂ rad ( B )and ( B, K ) is perfect, one can extend ψ to a ring homomorphism ψ : A h F ; f , f i → B satisfying φ = ψ ◦ Φ h F ; f ,f i as ring homomorphisms. Since ¯ G ( x ) = ¯ φ ( f ( x )) , ¯ G =¯ φ ( f ( x )), the following diagram is commutative A h F ; f , f i ψ −→ B ↓ ↓ A/I ¯ φ −→ B/K
Using the commutativity of this diagram and Lemma 2.1, we conclude that ψ : ( A h F ; f , f i , I h F ; f , f i ) → ( B, K )is, in fact, a morphism of pairs. The uniqueness of ψ follows from the factthat ( B, K ) is a UFP by Proposition 3.1.
Let (
A, I ) be a perfect pair. A morphism φ : ( A, I ) → ( B, J ) of pairs is calleda simple left factorization extension (or simple
LF-extension for short) of(
A, I ) if φ = Φ h F ; f ,f i for some polynomials F ( x ) ∈ A [ x ], f ( x ) , f ( x ) ∈ ( A/I )[ x ] satisfying the conditions in Proposition 3.2. An LF-extension of(
A, I ) is a morphism φ : ( A, I ) → ( B, J ) of pairs which is obtained by a finitesequence of simple LF-extensions, that is, there are simple LF-extensions φ i : ( A i , I i ) → ( A i +1 , I i +1 ) , where i = 1 , ..., d, such that ( A , I ) = ( A, I ), ( A d +1 , I d +1 ) = ( B, J ) and φ = φ d ◦ φ d − ◦· · · ◦ φ . Obviously, the collection of all LF-extensions of a perfect pair (
A, I ) isa set which we denote by
LF ext ( A, I ). Given morphisms φ : ( A, I ) → ( B , J ) , and φ : ( A, I ) → ( B , J )in LF ext ( A, I ), we write φ ≤ φ if there exists a morphism ψ : ( B , J ) → ( B , J )of pairs such that φ = ψ ◦ φ . Clearly, the relation ≤ defines a partial orderon LF ext ( A ). Furthermore, we have the following result.12 emma 3.3. Let ( A, I ) be a perfect pair. (i) For all φ : ( A, I ) → ( B , J ) and φ : ( A, I ) → ( B , J ) in LF ext ( A ) , there exists at most one morphism ψ : ( B , J ) → ( B , J ) of pairs such that φ = ψ ◦ φ . (ii) The partial order ≤ on LF ext ( A ) is directed.Proof. Part (i) can be proved using induction and the fact that ( B , J ) isa UFP. To prove (ii), we first assume that φ : ( A, I ) → ( B , J ) is a simpleLF-extension. Therefore, φ = Φ h F ; f ,f i for some polynomials F ( x ) ∈ A [ x ], f ( x ) , f ( x ) ∈ ( A/I )[ x ] satisfying the conditions in Proposition 3.2. Let G ( x ) = φ ( F ( x )), g ( x ) = ¯ φ ( F ( x )) and g ( x ) = ¯ φ ( F ( x )). It is easy to seethat the polynomials G ( x ) , g ( x ) , g ( x ) satisfy the conditions in Proposition3.2, giving rise to a morphismΦ h G ; g ,g i : ( B , J ) → ( B h G ; g , g i , I h G ; g , g i )of pairs. ClearlyΦ h G ; g ,g i ◦ φ : ( A, I ) → ( B h G ; g , g i , I h G ; g , g i )is an LF-extension. By the universal property of φ = Φ h F ; f ,f i , we see thatthere exists a morphism ψ : ( B , J ) → ( B h G ; g , g i , I h G ; g , g i )such that Φ h G ; g ,g i ◦ φ = ψ ◦ φ . It follows that φ ≤ Φ h G ; g ,g i ◦ φ and φ ≤ Φ h G ; g ,g i ◦ φ . The general case is proved by induction.
In this subsection, we prove the following result concerning the concept ofHenselization.
Theorem 3.4.
Let ( A, I ) be a perfect pair. Then, there exists a left Henselianpair ( A lh , I lh ) and a morphism φ lh : ( A, I ) → ( A lh , I lh ) of pairs having thefollowing universal property. For every morphism φ : ( A, I ) → ( B, J ) ofpairs from ( A, I ) to a left Henselian prefect pair ( B, J ) , there exists a uniquemorphism ψ : ( A lh , I lh ) → ( B, J ) of pairs such that φ = ψ ◦ φ lh .Proof. By Lemma 3.3, the direct limit ( A lh , I lh ) of elements in LF ext ( A )exists. Moreover, it is a perfect pair. We also have a canonical morphism φ lh : ( A, I ) → ( A lh , I lh )of pairs. The universal property of φ h follows from Proposition 3.2 and prop-erties of direct limits. So, it remains to show that ( A lh , I lh ) is left Henselian.13ince ( A lh , I lh ) is a direct limit of Jacobson pairs, it is a Jacobson pair. Leta monic polynomial F ( x ) ∈ A lh [ x ] be given such that ¯ F ( x ) = f ( x ) f ( x ) forsome left coprime monic polynomials f ( x ) , f ( x ) ∈ ( A lh /I lh )[ x ]. Since F ( x )has only finitely many (nonzero) coefficients, there exists an LF-extension φ : ( A, I ) → ( B, J ) such that F ( X ) ∈ B [ X ]. By Proposition 3.2, thepolynomial F ( X ) has a factorization F ( x ) = F ( x ) F ( x ) over B ( F ; f , f ),hence over A lh , such that F , F ∈ A lh [ X ] are monic, and ¯ F ( x ) = f ( x ),¯ F ( x ) = f ( x ). Since ( A lh , I lh ) is a perfect Jacobson ring, it is a UFP, seeProposition 3.1. Therefore, the factorization F ( x ) = F ( x ) F ( x ) is unique.It follows that ( A lh , I lh ) is a left Henselian pair, and we are done.The pair ( A lh , I lh ) is called the left Henselization of ( A, I ). It is easy tosee that the pair ( A lh , I lh ) is unique up to unique isomorphism. Similarly,one can show that every perfect pair ( A, I ) has a right Henselizaiton φ rh : ( A, I ) → ( A rh , I rh )satisfying the corresponding universal property. We note that if A/I is, inaddition, a commutative ring, then the right Henselization of (
A, I ) is alsothe left Henselization of (
A, I ), and vice versa.
A pair (
A, I ) is called commutative if A is a commutative ring. It is knownthat every commutative pair has a Henselization in the category P c of com-mutative pairs, see [6]. The Henselization of a commutative pair ( A, I )in P c is referred to as the commutative Henselization of ( A, I ), and is de-noted by ( A ch , I ch ). Obviously, the category P c is a full subcategory of P .The following result determines commutative Henselizations in terms of leftHenselizations. Proposition 3.5.
Let ( A, I ) be a commutative pair. Then, the ideal J generated by [ A lh , A lh ] is contained in I lh . Moreover, the morphism q ◦ φ lh : ( A, I ) → ( AJ lh , IJ lh ) , where q : ( A lh , I lh ) → ( A lh /J, I lh /J ) is the quotient morphism, is the com-mutative Henselization of ( A, I ) .Proof. Let ( A ch , I ch ) be the commutative Henselization of ( A, I ) and φ ch : ( A, I ) → ( A ch , I ch ) , be the corresponding morphism of pairs. Since ( A ch , I ch ) is left Henselian,there exists a unique morphism φ : ( A lh , I lh ) → ( A ch , I ch ) ,
14f pairs such that φ ch = φ ◦ φ lh . The fact that ( A ch , I ch ) is commutativeimplies that the ideal J generated by [ A lh , A lh ] is contained in the idealker( φ ) = φ − (0) ⊂ φ − ( I ch ) = I lh . Moreover, there exists a unique morphism ψ : ( AJ lh , IJ lh ) → ( A ch , I ch )such that φ = ψ ◦ q . Using the universal properties of φ lh and φ ch , one canverify that ψ is an isomorphism and the morphism q ◦ φ lh : ( A, I ) → ( AJ lh , IJ lh )is the commutative Henselization of ( A, I ). We conclude this article with a discussion of Henselizations of local rings.A ring A is called local if the set of all nonunits in A form an ideal. Everylocal ring A has a unique maximal ideal I . A pair ( A, I ) is called a localpair if A is a local ring and I is its maximal ideal. We note that any localpair is a Jacobson pair. Proposition 3.6.
The left (right) Henselization of any perfect local pair isa local pair.Proof.
Since the direct limit of local pairs is a local pair, it is enough toshow that any simple LF-extension of a perfect local pair is a local pair. LetΦ h F ; f ,f i : ( A, I ) → ( A h F ; f , f i , I h F ; f , f i )be a simple LF-extension where ( A, I ) is a local ring. Referring to thenotations used in Proposition 3.2, one can see that the ring homomorphism γ : R → A/I is a local homomorphism. Since
A/I is a local ring, Lemma2.5 implies that R is a local ring whose maximal ideal is J = γ − (0). Usingthe relation ( A h F ; f , f i , I h F ; f , f i ) = ̥ ( R, J ) , we conclude that ( A h F ; f , f i , I h F ; f , f i ) is a local pair, and we are done.15 eferences [1] Masood Aryapoor. Non-commutative henselian rings. Journal of Alge-bra , 322(6):2191–2198, 2009.[2] Masood Aryapoor. F-schemes. arXiv preprint arXiv:1001.1862 , 2010.[3] Gorˆo Azumaya. On maximally central algebras.
Nagoya MathematicalJournal , 2:119–150, 1951.[4] Paul Moritz. Cohn. Skew fields: theory of general division rings, 1995.[5] Silvio Greco. Henselization of a ring with respect to an ideal.
Transac-tions of the American Mathematical Society , 144:43–65, 1969.[6] J Lafon. Anneaux hens´eliens.
Bulletin de la Soci´et´e Math´ematique deFrance , 91:77–107, 1963.[7] Tsit-Yuen Lam.
A first course in noncommutative rings , volume 131.Springer Science & Business Media, 2013.[8] Masayoshi Nagata. Local rings.