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arXiv:0804.1221v1 [math.RA] 8 Apr 2008

STRONG CLEANNESS OF MATRIX RINGS OVER

COMMUTATIVE RINGS

FRANC ¸OIS COUCHOT

Abstract. Let R be a commutative local ring. It is proved that R is Henselian

if and only if each R-algebra which is a direct limit of module finite R-algebras

is strongly clean. So, the matrix ring Mn(R) is strongly clean for each integer

n > 0 if R is Henselian and we show that the converse holds if either the residue

class field of R is algebraically closed or R is an integrally closed domain or

R is a valuation ring. It is also shown that each R-algebra which is locally

a direct limit of module-finite algebras, is strongly clean if R is a π-regular

commutative ring.

As in [10] a ring R is called clean if each element of R is the sum of an idempotent

and a unit. In [8] Han and Nicholson proved that a ring R is clean if and only if

Mn(R) is clean for every integer n ≥ 1. It is easy to check that each local ring

is clean and consequently every matrix ring over a local ring is clean. On the

other hand a ring R is called strongly clean if each element of R is the sum

of an idempotent and a unit that commute. Recently, in [12], Chen and Wang

gave an example of a commutative local ring R with M2(R) not strongly clean.

This motivates the following interesting question: what are the commutative local

rings R for which Mn(R) is strongly clean for each integer n ≥ 1? In [4], Chen,

Yang and Zhou gave a complete characterization of commutative local rings R with

M2(R) strongly clean. So, from their results and their examples, it is reasonable

to conjecture that the Henselian rings are the only commutative local rings R with

Mn(R) strongly clean for each integer n ≥ 1. In this note we give a partial answer

to this problem. If R is Henselian then Mn(R) is strongly clean for each integer

n ≥ 1 and the converse holds if R is an integrally closed domain, a valuation ring

or if its residue class field is algebraically closed.

All rings in this paper are associative with unity. By [11, Chapitre I] a com-

mutative local ring R is said to be Henselian if each commutative module-finite

R-algebra is a finite product of local rings. It was G. Azumaya ([1]) who first

studied this property which was then developed by M. Nagata ([9]). The following

theorem gives a new characterization of Henselian rings.

Theorem 1. Let R be a commutative local ring. Then the following conditions are

equivalent:

(1) R is Henselian;

(2) For each R-algebra A which is a direct limit of module-finite algebras and

for each integer n ≥ 1, the matrix ring Mn(A) is strongly clean;

(3) Each R-algebra A which is a direct limit of module-finite algebras is clean.

2000 Mathematics Subject Classification. Primary 13H99, 16U99.

Key words and phrases. clean ring, strongly clean ring, local ring, Henselian ring, matrix ring,

valuation ring.

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2 FRANC ¸OIS COUCHOT

Proof. (1) ⇒ (2). Let A be a direct limit of module-finite R-algebras and

a ∈ Mn(A). Then R[a] is a commutative module-finite R-algebra. Since R is

Henselian, R[a] is a finite direct product of local rings. So R[a] is clean. Hence a is

a sum of an idempotent and a unit that commute.

It is obvious that (2) ⇒ (3).

(3) ⇒ (1). Let A be a commutative module-finite R-algebra and let J(A) be its

Jacobson radical. Since J(R)A ⊆ J(A), where J(R) is the Jacobson radical of R,

we deduce that A/J(A) is semisimple artinian. By [10, Propositions 1.8 and 1.5]

idempotents can be lifted modulo J(A). Hence A is semi-perfect. It follows that A

is a finite product of local rings, whence R is Henselian.

?

Let P be a ring property. We say that an algebra A over a commutative ring R

is locally P if AP satisfies P for each maximal ideal P of R.

Corollary 2. Let R be a commutative ring. Then the following conditions are

equivalent:

(1) R is clean and locally Henselian;

(2) For each R-algebra A which is locally a direct limit of module-finite algebras

and for each integer n ≥ 1, Mn(A) is strongly clean;

(3) Each R-algebra A which is locally a direct limit of module-finite algebras is

clean.

Proof. (1) ⇒ (2). Let A be an R-algebra which is locally a direct limit of

module-finite algebras and a ∈ Mn(A). Consider the following polynomial equa-

tions: E + U = a, E2= E, UV = 1, V U = 1, EU = UE. By Theorem 1 these

equations have a solution in Mn(AP), for each maximal ideal P of R. So, by [5,

Theorem I.1] they have a solution in Mn(A) too.

It is obvious that (2) ⇒ (3).

(3) ⇒ (1). Let P be a maximal ideal of R and let A be a module-finite RP-

algebra. Since R is clean, the natural map R → RP is surjective by [5, Theorem

I.1 and Proposition III.1]. So A is a module-finite R-algebra. It follows that A is

clean. By Theorem 1 RP is Henselian.

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A ring R is said to be strongly π-regular if, for each r ∈ R, there exist s ∈ R

and an integer q ≥ 1 such that rq= rq+1s.

Corollary 3. Let R be a strongly π-regular commutative ring. Then, for each

R-algebra A which is locally a direct limit of module-finite algebras and for each

integer n ≥ 1, the matrix ring Mn(A) is strongly clean.

Proof. It is known that R is clean and that each prime ideal is maximal. So, for

every maximal P, PRP is a nilideal of RP. Hence RP is Henselian. We conclude

by Corollary 2.

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By [6, Th´ eor` eme 1] each strongly π-regular R satisfies the following condition:

for each r ∈ R, there exist s ∈ R and an integer q ≥ 1 such that rq= srq+1.

Moreover, by [3, Proposition 2.6.iii)] each strongly π-regular ring is strongly clean.

So, Corollary 3 is also a consequence of the following proposition. (Probably, this

proposition is already known).

Proposition 4. Let R be a strongly π-regular commutative ring. Then, for each

R-algebra A which is locally a direct limit of module-finite algebras and for each

integer n ≥ 1, the matrix ring Mn(A) is strongly π-regular.

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STRONG CLEANNESS OF MATRIX RINGS3

Proof. Let S = Mn(A) and s ∈ S. Then R[s] is locally a module-finite algebra.

It is easy to prove that each prime ideal of R[s] is maximal. Consequently R[s] is

strongly π-regular. So, S is strongly π-regular too.

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The following lemma will be useful in the sequel.

Lemma 5. Let R be a commutative local ring with maximal ideal P. Let n be

an integer > 1 such that Mn(R) is strongly clean. Let f be a monic polynomial of

degree n with coefficients in R such that f(0) ∈ P and f(a) ∈ P for some a ∈ R\P.

Then f is reducible.

Proof. Let A ∈ Mn(R) such that its characteristic polynomial is f, i.e. f =

det(XIn− A), where In is the unit element of Mn(R). Then A = E + U where

E is idempotent, U is invertible and EU = UE. First we assume that a = 1. So,

0 and 1 are eigenvalues of A the reduction of A modulo P. Consequently A and

A − Inare not invertible. It follows that E ?= Inand E ?= 0n,nwhere 0p,q is the

p × q matrix whose coefficients are 0. Let F be a free R-module of rank n and let

ǫ be the endomorphism of F for which E is the matrix associated with respect to

some basis. Then F = Im ǫ ⊕ Ker ǫ. Moreover Im ǫ and Ker ǫ are free because R

is local. Consequently there exists a n × n invertible matrix Q such that:

QEQ−1= B =

?Ip

0q,p

0p,q

0q,q

?

where p is an integer such that 0 < p < n and q = n−p. Since E and A commute,

then B and QAQ−1commute too. So, QAQ−1is of the form:

QAQ−1=

?C

0q,p

0p,q

D

?

where C is a p×p matrix and D is a q×q matrix. We deduce that f is the product of

the characteristic polynomial g of C with the characteristic polynomial h of D. Let

us observe that (C−Ip) and D are invertible. So, g(1) / ∈ P, h(0) / ∈ P, g(0) ∈ P and

h(1) ∈ P. Now suppose that a ?= 1. Then a−nf(X) = g(Y ) where Y = a−1X and

g is a monic polynomial of degree n. We easily check that g(1) ∈ P and g(0) ∈ P.

It follows that g is reducible, whence f is reducible too.

?

A commutative ring R is a valuation ring (respectively arithmetic) if its

lattice of ideals is totally ordered by inclusion (respectively distributive).

Theorem 6. Let R be a local commutative ring with maximal ideal P and with

residue class field k. Consider the following two conditions:

(1) R is Henselian;

(2) the matrix ring Mn(R) is strongly clean ∀n ∈ N∗.

Then (1) ⇒ (2) and the converse holds if R satisfies one of the following properties:

(a) k is algebraically closed;

(b) R is an integrally closed domain;

(c) R is a valuation ring.

Proof. By Theorem 1 it remains to prove that (2) implies (1) when one of

(a), (b) or (c) is valid. We will use [2, Theorem 1.4] and [7, Theorem II.7.3.(iv)].

Consider the polynomial f = Xn+ cn−1Xn−1+ ··· + c1X + c0and assume that

∃m, 1 ≤ m < n such that cm / ∈ P and ci∈ P, ∀i < m. Since c0∈ P, we see that

f(0) ∈ P.

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Hence, if k is algebraically closed, ∃a ∈ R \ P such that f(a) ∈ P. By Lemma 5

f is reducible. So, by [2, Theorem 1.4] R is Henselian.

If R is an integrally closed domain, we take m = n−1 for proving the condition

(iv) of [7, Theorem II.7.3]. In this case f(−cn−1) ∈ P. By Lemma 5 (possibly

applied several times) f satisfies the condition (iv) of [7, Theorem II.7.3]. Hence R

is Henselian.

Assume that R is a valuation ring. Let N be the nilradical of R and let R′=

R/N. We know that R is Henselian if and only if R′is Henselian too. For each

n ∈ N∗, Mn(R′) is strongly clean. Since R′is a valuation domain, R′is integrally

closed. It follows that R′and R are Henselian.

?

Corollary 7. Let R be an arithmetic commutative ring. Then the following con-

ditions are equivalent:

(1) R is clean and locally Henselian;

(2) the matrix ring Mn(R) is strongly clean ∀n ∈ N∗.

Proof. By Corollary 2 it remains to show (2) ⇒ (1). Let P be a maximal ideal

of R. Since R is clean the natural map R → RP is surjective by [5, Theorem I.1

and Proposition III.1]. So, Mn(RP) is strongly clean ∀n ∈ N∗. Theorem 6 can be

applied because RP is a valuation ring. We conclude that RP is Henselian.

?

The following generalization of [4, Theorem 8] holds even if the properties

(a),(b),(c) of Theorem 6 are not satisfied.

Theorem 8. Let R be a local commutative ring with maximal ideal P and with

residue class field k. Let p be an integer such that 2 ≤ p ≤ 5. Then the following

conditions are equivalent:

(1) Mn(R) is strongly clean ∀n, 2 ≤ n ≤ p;

(2) each monic polynomial f of degree n, 2 ≤ n ≤ p, for which f(0) ∈ P and

f(1) ∈ P, is reducible.

Proof. By Lemma 5 it remains to prove that (2) ⇒ (1). Let A ∈ Mn(R).

We denote by f the characteristic polynomial of A. If A is invertible then A =

0n,n+ A. If A − Inis invertible then A = In+ (A − In). So, we may assume that

A and (A − In) are not invertible. It follows that f(0) ∈ P and f(1) ∈ P. Then,

f = gh where g and h are monic polynomials of degree ≥ 1. We may assume that

g(0) ∈ P, g(1) / ∈ P, h(0) / ∈ P and h(1) ∈ P (possibly by applying condition (2)

several times). We denote by¯f, ¯ g,¯h the images of f, g, h by the natural map

R[X] → k[X]. If ¯ g and¯h have a common factor of degree ≥ 1 then this factor

is of degree 1 because n ≤ 5. In this case ∃a ∈ R \ P such that g(a) ∈ P and

h(a) ∈ P. As in the proof of Lemma 5 we show that g is reducible. Hence, after

changing g and h, we get that ¯ g and¯h have no common divisor of degree ≥ 1. It

follows that there exist two polynomials u and v with coefficients in R such that

¯ u¯ g + ¯ v¯h = 1. Since PR[A] is contained in the Jacobson radical of R[A], we may

assume that u(A)g(A)+v(A)h(A) = In. We put e = vh. Then we easily check that

e(A) is idempotent. It remains to show that (A − e(A)) is invertible. It is enough

to prove that (¯ A − ¯ e(¯A)) is invertible because PMn(R) is the Jacobson radical of

Mn(R). Let V be a vector space of dimension n over k and let B be a basis of

V . Let α be the endomorphism of V for which¯A is the matrix associated with

respect to B. We put ǫ = ¯ e(α). Since V has finite dimension, it is sufficient to

show that (α − ǫ) is injective. Let w ∈ V such that α(w) = ǫ(w). It follows that

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STRONG CLEANNESS OF MATRIX RINGS5

α(ǫ(w)) = ǫ(α(w)) = ǫ2(w) = ǫ(w). Since ¯ e is divisible by (X −¯1) we get that

ǫ(w) = 0. So, α(w) = 0. We deduce that ǫ(w) = w because ¯ e −¯1 is divisible by X.

Hence w = 0.

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Laboratoire de Math´ ematiques Nicolas Oresme, CNRS UMR 6139, D´ epartement de

math´ ematiques et m´ ecanique, 14032 Caen cedex, France

E-mail address: couchot@math.unicaen.fr

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