Distributive Noetherian Centrally Essential Rings
aa r X i v : . [ m a t h . R A ] A ug Distributive Noetherian Centrally Essential Rings
Victor Markov
Lomonosov Moscow State Universitye-mail: [email protected]
Askar Tuganbaev
National Research University ”MPEI”Lomonosov Moscow State Universitye-mail: [email protected]
Abstract.
It is proved that a ring A is a right or left Noetherian, rightdistributive centrally essential ring if and only if A = A × · · · × A n ,where each of the rings A i is either a commutative Dedekind domain ora uniserial Artinian centrally essential (not necessarily commutative)ring.V.T.Markov is supported by the Russian Foundation for Basic Re-search, project 17-01-00895-A. A.A.Tuganbaev is supported by RussianScientific Foundation, project 16-11-10013. Key words: centrally essential ring, distributive ring, uniserial ring,Noetherian ring, Artinian ring.
MSC2010 datebase 16D25; 16D10 Introduction
All rings considered are associative and contain a non-zero identityelement. Writing expressions of the form “ A is an Artinian ring” meanthat the both modules A A and A A are Artinian.A ring A is said to be centrally essential if for any non-zero element a ∈ A , there are two non-zero central elements c, d ∈ A with ac = d ,i.e., the module A C is an essential extension of the module C C , where C is the center of A .It is clear that all commutative rings are centrally essential. If Z isthe field of order 2 and Q is the quaternion group of order 8, thenthe group ring Z [ Q ] is an example of a non-commutative centrallyessential ring [7]; in addition, Z [ Q ] is a finite local ring.A module is said to be distributive if the lattice of all its submodulesis distributive. A module is said to be uniserial if any two its submod-ules are comparable with respect to inclusion. It is clear that every Centrally essential rings are studied, for example, in [7], [8], [9]. uniserial module is distributive. The ring of integers Z is an exampleof a commutative distributive non-uniserial ring.This paper is a continuation of [9] which is a continuation of [8]. InTheorem 1.1, we recall some results of these papers. Theorem 1.1; [9] . 1)
Any finite, right uniserial, centrally essentialring is commutative and there are non-commutative uniserial Artiniancentrally essential rings; see [8]. A centrally essential ring R is a right uniserial, right Noetherianring if and only if R is a commutative discrete valuation domain or a(not necessarily commutative) uniserial Artinian ring.; see [9].In connection with Theorem 1.1, we prove Theorem 1.2 which is themain result of this paper. Theorem 1.2. 1) If A is a right distributive centrally essential finitering, then A = A × · · · × A n , where A i is a commutative uniserial finitering, i = 1 , . . . , n . In particular, A is a commutative distributive ring.Conversely, any such a direct product is a commutative distributivefinite ring. If A is a right distributive, right or left Noetherian, centrally essen-tial ring, then A = A × · · · × A n , where A i is either a (not necessarilycommutative) uniserial Artinian ring or a commutative Dedekind do-main, i = 1 , . . . , n . In particular, A is a distributive Noetherian ring.Conversely, any such a direct product is a distributive Noetherianring which is not necessarily commutative, since there exist non-commutative uniserial Artinian centrally essential rings; see Theorem1.1(1).The proof of the Theorem 1.2 is given in the next section and is basedon several assertions, some of which are of independent interest.We give some notation and definitions.Let M be a module. The module M is said to be uniform if any two itsnon-zero submodules have the non-zero intersection. A module M iscalled an essential extension of some its submodule X if X ∩ Y = 0 forany non-zero submodule Y in M . For any ring A , we denote by J ( A ), A ∗ and C ( A ) the Jacobson radical, the group of invertible elementsand the center of A , respectively.The left annihilator of an arbitrary subset S of the ring A is ℓ A ( S ) = { a ∈ A | aS = 0 } . The right annihilator r ( S ) of the set S are similarlydefined. We denote by [ a, b ] = ab − ba for the commutator of theelements a, b of an arbitrary ring. Let A be a ring and B a proper ideal of A . A ring A is called a domain if A does not have non-zero zero-divisors. The ideal B is said to be completely prime if the factor ring A/B is a domain.Other ring-theoretical notions and notations can be found in [5, 6, 11].2.
The proof of Theorem 1.2
Lemma 2.1.
Let A be a centrally essential ring such that the set B of all left zero-divisors of the ring A is an ideal of the ring A . Then B coincides with the set B ′ of all right zero-divisors of the ring A and A/B is a commutative domain. In addition, if the ideal B is nilpotentof nilpotence index n + 1, then ideal B n is contained in the center ofthe ring A . Proof.
Let a, a ′ ∈ A \ B .If aa ′ ∈ B , then there exists a non-zero element x ∈ A with aa ′ x = 0.Since a ′ / ∈ B , we have a ′ x = 0. Since a / ∈ B , we have aa ′ x = 0. This isa contradiction. Therefore, A/B is a domain.Since A is a centrally essential ring, there exist two non-zero centralelements x, y of A with bx = y . Then [ a, b ] x = [ a, bx ] = 0, i.e., [ a, b ] isa left zero-divisor. Consequently, [ a, b ] ∈ B and ring A/B is commuta-tive.Let a , a be two non-zero elements of the centrally essential ring A and a a = 0. There exist four non-zero central elements x , x , y , y of A such that a x = y , a x = y . Then y a = a y = a a x = 0.Therefore, the left zero-divisor a is a right zero-divisor. Similarly, theright zero-divisor a is a left zero-divisor. Therefore, B = B ′ .Now let the ideal B be nilpotent of nilpotence index n + 1. It remainsto prove that ab − ba = 0 for any elements a ∈ A and b ∈ B n .Let’s assume that ab − ba = 0 for some elements a ∈ A and b ∈ B n .Since A is a centrally essential ring, there exist central elements x, y ∈ A with bx = y = 0. Then abx = ay = ya = bxa = bax , ( ab − ba ) x = abx − bax = 0. Therefore, x is a central zero-divisor. Consequently, x ∈ B . Then y = bx ∈ B n +1 = 0. This is a contradiction. (cid:3) For convenience, we give the proof of the following well known Lemma2.2.
Lemma 2.2.
Let A be a commutative domain and there exists a non-zero finitely generated divisible torsion-free A -module M . Then A is afield. Proof.
We assume the contrary. Then A has a maximal ideal m = 0and M can be naturally turn into a non-zero finitely generated moduleover the local ring A m with radical J = m A m . Since the module M isdivisible, we have that M J ⊇ M m = M , and M = 0 by the Nakayamalemma. This is a contradiction. (cid:3) Proposition 2.3.
Let A be a right distributive indecomposable ringwith the maximum condition on right annihilators, which does notcontain an infinite direct sum of non-zero right ideals, and let B be theprime radical of the ring A . Suppose that A be not a right uniformdomain. Then: . The ideal B is a non-zero completely prime nilpotent ideal, B is the set of all left zero-divisors of the ring A , B is a essential rightideal and B = aB for any element a ∈ A \ P . . If the ring A is centrally essential and B is a finitely generatedright ideal of the ring A , then A is a uniserial Artinian ring with radical B . Proof. 2.3.1 . The assertion is proved in [11, The assertion 9.20(1)]. . By 2.2.1 and Lemma 2.1, B is a non-zero completely primenilpotent ideal, B coincides with the set of all left or right zero-divisorsof the ring A , A/B is a commutative domain and B = aB for anyelement a ∈ A \ B .Let n + 1 be the of nilpotence index of the ideal B . Then B n is a leftand right module over the commutative domain A/B . Let a ∈ A \ B .Since B = aB and B is an ideal, B n = aB n and B n is a divisibleleft A/B -module. By Lemma 2.1, the ideal B n is contained in thecenter of the ring A . Therefore, the divisible left A/B -module B n is adivisible right A/B -module. Since B is a finitely generated right idealwhich is an ideal, B n is a finitely generated right ideal. Therefore, thecommutative domain A/B has a non-zero finitely generated divisibleright module B n . By Lemma 2.1, any element of the set A \ B is nota right zero-divisor. Therefore, B n is a torsion-free right A/B -module.By Lemma 2.2,
A/B is a field. Therefore, A is a right distributivelocal ring with non-zero nilpotent Jacobson radical B . Then A is aright uniserial Artinian right ring [10]. Since A is a right uniserial,right Artinian, centrally essential ring, A is a uniserial Artinian ring[8, Corollary 2.4]. (cid:3) Corollary 2.4.
Let A be a right distributive Noetherian right centrallyessential indecomposable ring with non-zero prime radical B . Then A is a uniserial Artinian ring with radical B . Corollary 2.4 follows from Lemma 2.3.2.
Remark 2.5.
A semiprime ring A is centrally essential if and only if A is commutative [7, Proposition 3.3]. Consequently, a prime ring is acentrally essential if and only if A is a commutative domain. Remark 2.6.
It is well known that commutative domain A is a dis-tributive Noetherian ring if and only if A is a Dedekind domain; e.g.,see [1, Theorem 6] or [11, Theorem 9.16]. It follows from this propertyand Remark 2.5 that the right distributive, right or left Noetherian,centrally essential prime rings coincide with commutative Dedekinddomains. Such rings are distributive Noetherian rings. Proposition 2.7.
A ring A is a right distributive, right Noetheriancentrally essential ring if and only if A = A × · · · × A n , where A i isa uniserial Artinian centrally essential ring or commutative Dedekinddomain, i = 1 , . . . , n . Conversely, it is clear that in this case A is adistributive Noetherian ring.Proposition 2.7 follows from Corollary 2.4 and Remarks 2.5 and 2.6. Proposition 2.8.
A ring A is a right distributive, left Noetheriancentrally essential ring if and only if A = A × · · · × A n , where A i is auniserial Artinian centrally essential ring or a commutative Dedekinddomain, i = 1 , . . . , n . Conversely, it is clear that A is a distributiveNoetherian ring in this case. Proof.
In [11, Theorem 9.18(1)], it is proved that a ring A is a rightdistributive, left Noetherian ring if and only if A = A × · · · × A n ,where A i is an Artinian, right uniserial ring or right distributive, leftNoetherian domain, i = 1 , . . . , n . In addition, any right uniserial cen-trally essential Artinian ring is a left uniserial ring [8, Corollary 2.4].Now we use Remark 2.6. (cid:3) The right distributive finite rings coincide with finite directproducts of right uniserial finite rings [10]. Therefore, 1.2(1) followsfrom Theorem 1.1(1).
The assertion follows from Propositions 2.7 and 2.8.3.
Remarks
In [4], it is proved that a commutative ring A isa distributive semiprime ring if and only if every submodule of any flat A -module is a flat module. Therefore, it follows from Remark 2.5that the right or left distributive, centrally essential, semiprime ringscoincide with commutative rings over which every submodule of anyflat module is a flat module. In [1], it is proved that the commutative distribu-tive semiprime rings, not containing an infinite direct sum of non-zeroideals, coincide with finite direct products of commutative semihered-itary domains. Therefore, it follows from Remark 2.5 that the rightor left distributive, centrally essential, semiprime rings, not contain-ing an infinite direct sum of non-zero ideals, coincide with finite directproducts of commutative semihereditary domains. We recall the transfinitedefinition of the Krull dimension Kdim M of the module M , see [2].By definition, we assume that zero modules are of Krull dimension − α is an ordinal number > β have been defined for all ordinal numbers β < α and let M be a module with Kdim M = β .We say that the Krull dimension
Kdim M of the module M is α if forany infinite strictly descending chain M > M > . . . of submodules of M , there exists a positive integer n with Kdim( M n /M n +1 ) < α .If A is a ring and the Krull dimension Kdim A A of the module A A existsthen Kdim A A is called the right Krull dimension of the ring A . Every Noetherian module has Krull dimension; see [2].
Not every module has Krull dimension; for example, any di-rect sum of an infinite set of non-zero modules does not have Krulldimension.
It follows from Theorem 1.2(2) that every right distributive, right Noe-therian, centrally essential, indecomposable ring A is either a commu-tative domain or an Artinian ring. In this assertion, we cannot replacethe condition ” A is a right Noetherian ring” by the condition ” A isa ring with right Krull dimension”. Indeed, let D be a commutativeuniserial principal ideal domain which is not a field and let Q be the A right A -module X is said to be flat if for every left A -module Y and anysubmodule Y ′ in Y , a natural homomorphism X ⊗ A Y ′ → X ⊗ A Y of additivegroups is a monomorphism. A module is said to be semihereditary if all its finitely generated submodules areprojective modules. field of fractions of the domain D . For example, we can take the for-mal power series ring D = F [[ x ]] over a field F and the formal Laurentseries field Q = F (( x )). Let A be the trivial extension of the D - D -bimodule Q by D . It is directly verified that A is a commutative,uniserial, non-Artinian, non-prime ring of Krull dimension 1. It follows from Theorem 1.2(2) that every right distributive, right Noe-therian, centrally essential, indecomposable ring A is either a commu-tative domain or a uniserial ring. In this assertion, we cannot replacethe condition ” A is a right Noetherian ring” by the condition ” A is aring with right Krull dimension”. Indeed, let Z be the ring of integers, p be a prime integer, M be the quasi-cyclic p -group, and let A be thetrivial extension of the Z - Z -bimodule M by Z . It is directly verifiedthat A is a commutative distributive, non-uniserial, non-Artinian, non-prime, indecomposable ring of Krull dimension 1. In addition, A is aBezout ring in which any ideal is comparable with respect to inclusionwith the ideal Q . References [1] Camillo V., Distributive modules // J. Algebra. – 1975. – Vol. . – P.16–25.[2] Gordon R., Robson J.C. Krull dimension. – Mem. Amer. Math. Soc. – 1973. –no. 133. – P. 1–78.[3] Jelisiejew J., On commutativity of ideal extensions // Comm. Algebra. – 2016.– Vol.44