Study of a division-like property Robin KhanrB eranger Seguin October 25 2022

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Study of a division-like property

Robin Khanﬁr∗B´eranger Seguin†

October 25, 2022

To our friend Assil Fadle.

Abstract

We introduce a weak division-like property for noncommutative rings: a nontrivial ring is

fadelian if for all nonzero a, x there exist b, c such that x=ab +ca. We prove properties of

fadelian rings, and construct examples of such rings which are not division rings, as well as

non-Noetherian and non-Ore examples.

Keywords: Noncommutative rings ·Simple algebras ·Ore condition ·Ore extension

Mathematics Subject Classiﬁcation (MSC 2010): 16U20 ·13N10 ·12E15 ·16P40

1 Introduction

This article is a compilation of results obtained between 2017 and 2019 on questions of non-

commutative algebra. At the time, we were a group of students grouped under the informal hu-

moristic name of D´epartement de Math´ematiques Inapplicables. Among others, Maxime Ramzi

is to thank for his precious help.

All rings considered are nontrivial, unital, and not assumed to be commutative. Various

notions of weak inversibility have already been considered, notably (strong) von Neumann reg-

ularity [Neu36] and unit regularity. We introduce a new class of rings satisfying a weak form of

divisibility. They are the fadelian and weakly fadelian rings:

Deﬁnition 1.1. A ring Ris:

•fadelian if for any x∈Rand any nonzero a∈R, there exist b, c ∈Rsuch that:

x=ab +ca;

•weakly fadelian if for any nonzero a∈R, there exist b, c ∈Rsuch that:

1 = ab +ca.

We have the following implications:

Ris a division ring ⇒Ris fadelian ⇒Ris weakly fadelian.

A natural question is whether any of these implications are equivalences, and to construct

counterexamples when they are not.

•In Section 2, we prove that weakly fadelian rings are simple (Proposition 2.1) and integral

(Theorem 2.2), and that weakly fadelian Ore rings are fadelian (Theorem 2.6).

•In Section 3, we study diﬀerential algebras. We give conditions for these to yield fadelian

rings (Proposition 3.5 and Theorem 3.6). These conditions are satisﬁed for diﬀerentially

closed ﬁelds, so this gives the ﬁrst example of a fadelian ring which is not a division

ring. In Theorem 3.13 and Theorem 3.15, we transform this example into a countable

non-Noetherian fadelian ring.

∗Sorbonne Universit´e, Campus Pierre et Marie Curie, LPSM, Case courrier 158, 4, place Jussieu, 75252 Paris

Cedex 05, France. Email: robin.khanfir@sorbonne-universite.fr

†Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev´e, F-59000 Lille, France. Email:

beranger.seguin@ens.psl.eu

arXiv:2210.13078v1 [math.RA] 24 Oct 2022

•In Section 4, we study Laurent series on fadelian rings. They are themselves fadelian

(Theorem 4.3) and turn previous examples into an example of a non-Ore fadelian ring

(Theorem 4.4, Corollary 4.5).

The results are summarized in the following map:

Division Ring

Fadelian

Ore

Weakly Fadelian

Integral

Simple

Noetherian

Figure 1: Known (non)-implications between the diﬀerent notions. Only a generating set of arrows is

drawn. The blue arrows are of the form “Aand Bimply C”. The red crossed arrow is known not to be an

implication.

The main open conjecture is the following one:

Conjecture 1. There is a weakly fadelian ring which is not fadelian.

The authors of this document oﬀer a pizza from Golosino to anyone who proves or disproves

Conjecture 1.

2 Properties of weakly fadelian rings

In this section, Ris a weakly fadelian ring. We prove various properties of R. The proofs of

Theorem 2.2 and Theorem 2.6 have been formalized in the Lean proof assistant, cf Appendix A.

A ﬁrst remark is that if Ris commutative, it is a ﬁeld. This is generalized by the following

proposition:

Proposition 2.1. The ring Ris simple.

Proof. Let Ibe a nonzero two-sided ideal of R. Let a∈I\ {0}. Since Ris weakly fadelian,

there are b, c ∈Rsuch that 1 = ab +ca. We obtain 1 ∈Iand ﬁnally I=R.

We now prove the following result:

Theorem 2.2. The ring Ris integral.

Here, integral means “containing no zero divisors”, without requiring commutativity. The

proof uses two lemmas:

Lemma 2.3. Assume x, y ∈Rsatisfy xy =yx = 0. Then x2= 0 or y2= 0.

Proof. If x= 0, this is immediate. Otherwise, use weak fadelianity to write

1 = xb +cx

for some b, c ∈R. Then:

y2=y·1·y

=y(xb +cx)y

=yxby +ycxy

= (yx)by +yc(xy)

= 0.

Lemma 2.4. Assume x∈Rsatisﬁes x2= 0. Then x= 0.

Proof. Assume by contradiction that xis nonzero. Write:

1 = xb +cx (1)

for some b, c ∈R. Notice that:

cx =cx ·1 = cx(xb +cx) = c(x2)b+ (cx)2= (cx)2.

Similarly, we have xb = (xb)2. Since xb = 1 −cx, we know that xb and cx commute. Hence:

(xb)(cx) = (cx)(xb) = c(x2)b= 0.

By Lemma 2.3, we have (xb)2= 0 or (cx)2= 0, and thus xb = 0 or cx = 0. Assume for example

that xb = 0. Equation (1) becomes 1 = cx, so xis invertible, which contradicts x2= 0.

We ﬁnally prove Theorem 2.2:

Proof of Theorem 2.2. Let x, y ∈Rsuch that xy = 0. Then (yx)2=y(xy)x= 0, which implies

yx = 0 by Lemma 2.4. By Lemma 2.3, we deduce from xy =yx = 0 that either x2= 0 or

y2= 0. Applying Lemma 2.4 again, we see that either xor yis zero.

The Ore condition is a well-studied condition, equivalent to the existence of a ring of fractions

unique up to isomorphism [Ore31]. It is weaker than Noetherianity ([Gol58, Theorem 1], [GW04,

Corollary 6.7]). We recall the deﬁnition:

Deﬁnition 2.5. The integral ring Ris right (resp. left) Ore if any two nonzero right (resp.

left) ideals have a nonzero intersection.

The Ore condition interacts with fadelianity in the following way:

Theorem 2.6. If the weakly fadelian ring Ris right Ore, it is fadelian.

Proof. Assume Ris right Ore. Let x, a ∈R\ {0}. Since Ris right Ore, there exist nonzero

elements b, c ∈Rsuch that:

ab =xc.

We have ca 6= 0 by Theorem 2.2. Since Ris weakly fadelian, there exist k, k0∈Rsuch that:

1 = cak +k0ca

Finally:

x=x·1 = xcak +xk0ca =abak +xk0ca ∈aR +Ra.

This proves that Ris fadelian.

In Corollary 4.5, we will see that Theorem 2.6 is not an equivalence.

3 Fadelianity and diﬀerential algebras

In this section, we consider diﬀerential algebras. We give conditions (both necessary and suﬃ-

cient) for formal diﬀerential operator rings in the sense of [GW04, Chapter 2] to be fadelian.

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Studyofadivision-likepropertyRobinKhanr*BerangerSeguinOctober25,2022ToourfriendAssilFadle.AbstractWeintroduceaweakdivision-likepropertyfornoncommutativerings:anontrivialringisfadelianifforallnonzeroa;xthereexistb;csuchthatx=ab+ca.Weprovepropertiesoffadelianrings,andconstructexamplesofsuchringswhi...

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