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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2023; 63(4): 561-569

Published online December 31, 2023 https://doi.org/10.5666/KMJ.2023.63.4.561

### Certain Clean Decompositions for Matrices over Local Rings

Yosum Kurtulmaz, Handan Köse, Huanyin Chen*

Department of Mathematics, Bilkent University, Ankara, Turkey
e-mail : yosum@fen.bilkent.edu.tr

Department of Mathematics, Ahi Evran University, Kirşehir, Turkey
e-mail : handan.kose@ahievran.edu.tr

School of Big Data, Fuzhou University of International Studies and Trade, Fuzhou 350202, China
e-mail : huanyinchenfz@163.com

Received: December 15, 2022; Revised: July 21, 2023; Accepted: July 25, 2023

### Abstract

An element aR is strongly rad-clean provided that there exists an idempotent eR such that aeU(R), ae=ea and eaeJ(eRe). In this article, we completely determine when a 2×2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.

Keywords: Strongly clean matrix, Strongly rad-clean matrix, Local ring, Power-series

### 1. Introduction

An element a ∈ R is strongly clean provided that it is the sum of an idempotent and a unit that commutes. A ring R is strongly clean provided that every element in R is strongly clean. A ring R is local if it has only one maximal right ideal. As is well known, a ring R is local if and only if for any x∈ R, x or 1-x is invertible. Strongly clean matrices over commutative local rings was extensively studied by many authors from very different view points (see [1, 2, 3, 5, 6, 7, 8, 9, 10, 11]). Recently, a related cleanness of triangular matrix rings over abelian rings was studied by Diesl et al. (see [9]).

Following Diesl, we say that a ∈ R is strongly rad-clean provided that there exists an idempotent e ∈ R such that aeU(R),ae=ea and eaeJ(eRe) (see [9]). A ring R is strongly rad-clean provided that every element in R is strongly rad-clean. Strongly rad-clean rings form a natural subclass of strongly clean rings which have stable range one (see [4]). Let M be a right R-module, and let φendR(M). Then we include a relevant diagram to reinforce the theme of direct sum decompositions:

M=ABφφM=AB

If such diagram holds we call this is an AB-decomposition for φ. It turns out by [2, Lemma 40] that φ is strongly π-regular if and only if there is an AB-decomposition with φ|BN(end(B)) (the set of nilpotent elements).

Further, φ is strongly rad-clean if and only if there is an AB-decomposition with φ|BJ(end(B)) (the Jacobson radical of end(B)). Thus, strong rad-cleanness can be seen as a natural extension of strong π-regularity. In [2, Theorem 12], the authors gave a criterion to characterize when a square matrix over a commutative local ring is strongly clean. We extend this result to strongly rad-clean matrices over a commutative local ring. We completely determine when a 2×2 matrix over a commutative local ring has such clean decomposition related to its Jacobson radical. Application to the matrices over power-series is also studied.

Throughout, all rings are commutative with an identity and all modules are unitary left modules. Let M be a left R-module. We denote the endomorphism ring of M by end(M) and the automorphism ring of M by aut(M), respectively. The characteristic polynomial of A is the polynomial χ(A)=det(tInA). We always use J(R) to denote the Jacobson radical and U(R) is the set of invertible elements of a ring R. M2(R) stands for the ring of all 2×2 matrices over R, and GL2(R) denotes the 2-dimensional general linear group of R.

### 2. Main Results

In this section, we study the structure of strongly rad-clean elements in various situations related to ordinary ring extensions which have roles in ring theory. We start with a well known characterization of strongly rad-clean element in the endomorphism ring of a module M.

Lemma 2.1. Let E=end(RM), and let αE. Then the following are equivalent:

• (1) αE is strongly rad-clean.

• (2) There exists a direct sum decomposition M=PQ where P and Q are α-invariant, and α|Paut(P) and α|QJ(end(Q)).

Proof. See [4, Proposition 4.1.2]. Lemma 2.2. Let R be a ring, let M be a left R-module. Suppose that x,y,a,bend(RM) such that xa+yb=1M,xy=yx=0,ay=ya and xb=bx. Then M=ker(x)ker(y) as left R-modules.

Proof. See [2, Lemma 11].

A commutative ring R is projective-free if every finitely generated projective R-module is free. Evidently, every commutative local ring is projective-free. We now derive

Lemma 2.3. Let R be projective-free. Then AM2(R) is strongly rad-clean if and only if AGL2(R), or AJ(M2(R)), or A is similar to diag(α,β) with αJ(R) and βU(R).

Proof. Write A=E+U,E2=E,UGL2(R),EA=AEJ(M2(R)). Since R is projective-free, there exists PGLn(R) such that PEP1=diag(0,0),diag(1,1) or diag(1,0). Then (i) PAP1=PUP1; hence, AGL2(R), (ii) (PAP1)diag(1,1)=diag(1,1)(PAP1)J(M2(R), and so AJ(M2(R)). (3) (PAP1)diag(1,0)=diag(1,0)(PAP1)J(M2(R) and PAP1diag(1,0)GL2(R). Hence, PAP1=abcd with aJ(R),b=c=0 and dUR). Therefore A is similar to diag(α,β) with αJ(R) and βU(R).

If AGL2(R) or AJ(M2(R)), then A is strongly rad-clean. We now assume that A is similar to diag(α,β) with αJ(R) and βU(R). Then A is similar to 1000+α100β where

α1 0 0 βGL2(R), α 0 0 β 1 0 0 0J(M2(R)) α1 0 0 β 1 0 0 0= 1 0 0 0 α1 0 0 β.

Therefore AM2(R) is strongly rad-clean.

Theorem 2.4. Let R be projective-free. Then AM2(R) is strongly rad-clean if and only if

• (1) AGL2(R)), or

• (2) AJ(M2(R)), or

• (3) x2=tr(A)xdetA has roots αU(R),βJ(R).

Proof. By Lemma 2.3, AGL2(R), or AJ(M2(R)), or A is similar to a matrix α00β, where αJ(R) and βU(R). Then χ(A)=(xα)(xβ) has roots αU(R),βJ(R).

If (1) or (2) holds, then AM2(R) is strongly rad-clean. If (3) holds, we assume that χ(A)=(tα)(tβ). Choose X=AαI2 and Y=AβI2. Then

X(βα)1I2Y(βα)1I2=I2,XY=YX=0,X(βα)1I2=(βα)1I2X,(βα)1I2Y=Y(βα)1I2.

By virtue of Lemma 2.2, we have 2R=ker(X)ker(Y). For any xker(X), we have (x)AX=(x)XA=0, and so (x)Aker(X). Then ker(X) is A-invariant. Similarly, ker(Y) is A-invariant. For any xker(X), we have 0=(x)X=(x)(AαI2); hence, (x)A=(x)αI2. By hypothesis, we have A|ker(X)J(end(ker(X))). For any yker(Y), we prove that

0=(y)Y=(y)(AβI2).

This implies that (y)A=(y)(βI2). Obviously, A|ker(Y)aut(ker(Y)). Therefore AM2(R) is strongly rad-clean by Lemma 2.1.We have accumulated all the information necessary to prove the following.

Theorem 2.5. Let R be a commutative local ring, and let AM2(R). Then the following are equivalent:

• (1) AM2(R) is strongly rad-clean.

• (2) AGL2(R) or AJ(M2(R)), or trAU(R) and the quadratic equation x2+x=detAtr2A has a root in J(R).

• (3) AGL2(R) or AJ(M2(R)), or trAU(R),detAJ(R) and the quadratic equation x2+x=detAtr2A4detA is solvable.

Proof. (1)(2) Assume that AGL2(R) and AJ(M2(R)). By virtue of Theorem 2.4, trAU(R) and the characteristic polynomial χ(A) has a root in J(R) and a root in U(R). According to Lemma 2.3, A is similar to λ00μ, where λJ(R),μU(R). Clearly, y2(λ+μ)y+λμ=0 has a root λ in J(R).

Hence so does the equation

(λ+μ)1y2y=(λ+μ)1λμ.

Set z=(λ+μ)1y. Then

(λ+μ)z2(λ+μ)z=(λ+μ)1λμ.

That is, z2z=(λ+μ)2λμ. Consequently, z2z=detAtr2A has a root in J(R). Let x=-z. Then x2+x=detAtr2A has a root in J(R).

(2)(3) By hypothesis, we prove that the equation y2y=detAtr2A has a root aJ(R). Assume that trA∈ U(R). Then (a(2a1)1)2(a(2a1)1)=detAtr2A(4(a2a)+1)=detAtr2A(4(trA)2detA+1)=detAtr2A4detA. Therefore the equation y2y=detAtr2A4detA is solvable. Let x=-y. Then x2+x=detAtr2A4detA is solvable.

(3)(1) Suppose AGL2(R) and AJ(M2(R)). Then trAU(R),detAJ(R) and the equation x2+x=detAtr2A4detA has a root. Let y=-x. Then y2ydetAtr2A4detA has a root aR. Clearly, b:=1aR is a root of this equation. As a2aJ(R), we see that either aJ(R) or 1aJ(R). Thus, 2a1=12(1a)U(R). It is easy to verify that (a(2a1)1trA)2trA(a(2a1)1trA)+detA=tr2A(a2a)4(a2a)+1+detA=0. Thus the equation y2trAy+detA=0 has roots a(2a1)1trA and b(2b1)1trA. Since ab∈ J(R), we see that a+b=1 and either aJ(R) or bJ(R). Therefore y2trAy+detA=0 has a root in U(R) and a root in J(R). Since R is a commutative local ring, it is projective-free. By virtue of Theorem 2.4, we obtain the result.

Corollary 2.6. Let R be a commutative local ring, and let AM2(R). Then the following are equivalent:

• (1) AM2(R) is strongly clean.

• (2) I2AGL2(R) or AM2(R) is strongly rad-clean.

Proof. (2)(1) is trivial.

(1)(2) In view of [3, Corollary 16.4.33], AGL2(R), or I2AGL2(R) or trAU(R),detAJ(R) and the quadratic equation

x2x=detAtr2A4detA is solvable. Hence x2+x=detAtr2A4detA is solvable. According to Theorem 2.5, we complete the proof.

Corollary 2.7. Let R be a commutative local ring. If 12R, then the following are equivalent:

• (1) AM2(R) is strongly rad-clean.

• (2) AGL2(R) or AJ(M2(R)), or trAU(R),detAJ(R) and tr2A4detA is square.

Proof. (1)(2) According to Theorem 2.5, AGL2(R) or AJ(M2(R)), or trAU(R),detAJ(R) and the quadratic equation x2x=detAtr2A4detA is solvable. If aR is the root of the equation, then (2a1)2=4(a2a)+1=tr2Atr2A4detAU(R). As in the proof of Theorem 2.5 , 2a1U(R). Therefore tr2A4detA=(trA(2a1)1)2.

(2)(1) If trAU(R),detAJ(R) and tr2A4detA=u2 for some uR, then uU(R) and the equation x2+x=detAtr2A4detA has a root 12u1(trA+u). By virtue of Theorem 2.5, AM2(R) is strongly rad-clean.

Every strongly rad-clean matrix over a ring is strongly clean. But there exist strongly clean matrices over a commutative local ring which is not strongly rad-clean as the following shows.

Example 2.8. Let R=4, and let A=2302M2(R). R is a commutative local ring. Then A=1001+1301 is a strongly clean decomposition. Thus AM2(R) is strongly clean. If AM2(R) is strongly rad-clean, there exist an idempotent EM2(R) and an invertible UM2(R) such that A=E+U, EA=AE and EAEJ(M2(R)). Hence, AU=A(AE)=(AE)A=UA, and then E=AUGL2(R) as A4=0. This implies that E=I2, and so EAE=AJ(M2(R)), as J(R)=2R. This gives a contradiction. Therefore AM2(R) is not strongly rad-clean.

Following Cui and Chen, an element aR is quaspolar if there exists an idempotent ecomm(a) such that a+eU(R) and aeRqnil (see [6]). Obviously, A is strongly J-cleanA is strongly rad-cleanA is quasipolar. But the converses are not true, as the following shows:

Example 2.9. (1) Let R be a commutative local ring and A=1110 be in M2(R). Since AGL2(R), by Lemma 2.3, it is strongly rad-clean but is not strongly J-clean, as I2AJ(M2(R)).

(2) Let R=(3) and A=2111. Then trA=3J(R) and detA=3J(R). Hence A is quasipolar by [5, Theorem 2.6]. Note that trAU(R), AGL2(R) and AJ(M2(R)). Thus, A is not strongly rad-clean, in terms of Corollary 2.7.

Set B12(a)=1a01 and B21(a)=10a1. We now derive

Theorem 2.10. Let R be a commutative local ring. Then the following are equivalent:

• (1) Every AM2(R) with invertible trace is strongly rad-clean.

• (2) For any λJ(R),μU(R), the quadratic equationx2=μx+λ is solvable.

Proof. (1)(2) Let λJ(R),μU(R). Choose A=0λ1μ. Then AM2(R) is strongly rad clean. Obviously, AGL2(R) and AJ(M2(R)). In view of Theorem 2.4, we see that the quadratic equation x2=μx+λ is solvable.

(2)(1) Let A=abcd with tr(A)U(R).

Case I. cU(R). Then

diag(c,1)B12(ac1)AB12(ac1)diag(c1,1)=0λ1μ

for some λ,μR. If λU(R), then AGL2(R), and so it is strongly rad-clean. If λJ(R), then μU(R). Then A is strongly rad-clean by Theorem 2.5.

Case II. bU(R). Then

0110A0110=dcba,

and the result follows from Case I.

Case III. c,bJ(R),adU(R). Then

B21(1)AB21(1)=a+bbca+dbdb

where ad+bcU(R); hence the result follows from Case I.

Case IV. c,bJ(R),a,dU(R). Then

B21(ca1)A=ab0dca1b;

hence, AGL2(R).

Case V. c,b,a,dJ(R). Then AJ(M2(R)), and so tr(A)J(R), a contradiction.

Therefore AM2(R) with invertible trace is strongly rad-clean.

Example 2.11. Let R=4. Then R is a commutative local ring. For any λJ(R),μU(R), we directly check that the quadratic equation x2=μx+λ is solvable. Applying Theorem 2.10, every 2×2 matrix over R with invertible trace is strongly rad-clean. In this case, M2(R) is not strongly rad-clean.

Example 2.12. Let R=^2 be the ring of 2-adic integers. Then every 2×2 matrix with invertible trace is strongly rad-clean.

Proof. Obviously, R is a commutative local ring. Let λJ(R),μU(R). Then 0λ1μM2(R) is strongly clean, by [5, Theorem 3.3]. Clearly, det(A)=λJ(R). As R/J(R)2, we see that μ1+J(R), and then det(AI2)=1λμJ(R). In light of [5, Lemma 3.1], the equation x2=μx+λ is solvable. This completes the proof, by Theorem 2.10.

We note that matrix with non-invertible trace over commutative local rings maybe not strongly rad-clean. For instance, A=1111M2(^2) is not strongly rad-clean.

### 3. Applications

We now apply our preceding results and investigate strongly rad-clean matrices over power series over commutative local rings.

Lemma 3.1. Let R be a commutative ring, and let A(x1,,xn)M2(R[[x1,,xn]]). Then the following hold:

• (1) A(x1,,xn)GL2(R[[x1,,xn]]) if and only if A(0,,0)GL2(R).

• (2) A(x1,,xn)J(M2(R[[x1,,xn]])) if and only if A(0,,0)J(M2(R)).

Proof. (1) We suffice to prove for n=1. If A(x1)GL2(R[[x1]]), it is easy to verify that A(0)GL2(R). Conversely, assume that A(0)GL2(R). Write

A(x1)= i=0aix1i i=0bix1i i=0cix1i i=0dix1i,

where A(0)=a0b0c0d0. We note that the determinant of A(x1) is a0d0c0b0+x1f(x1), which is a unit plus an element of the radical of R[[x1]]. Thus, A(x1)GL2(R[[x1]]), as required.

(2) It is immediate from (1).

Theorem 3.2. Let R be a commutative local ring, and let A(x1,,xn)M2(R[[x1,,xn]]). Then the following are equivalent:

• (1) A(x1,,xn)M2(R[[x1,,xn]]) is strongly rad-clean.

• (2) A(x1,,xn)M2(R[[x1,,xn]]/(x1m1xnmn)) is strongly rad-clean.

• (3) A(0,,0)M2(R) is strongly rad-clean.

Proof. (1)(2) and (2)(3) are obvious.

(3)(1) It will suffice to prove for n=1. Set x=x1. Clearly, R[[x]] is a commutative local ring. Since A(0) is strongly clean in M2(R), it follows from Theorem 2.4 that A(0)GL2(R), or A(0)J(M2(R)), or χ(A(0)) has a root αJ(R) and a root βU(R). If A(0)GL2(R) or A(0)J(M2(R)), in view of Lemma 3.1, A(x)GL2(R[[x]]) or A(x)J(M2(R[[x]])). Hence, A(x)M2(R[[x]]) is strongly rad-clean. Thus, we may assume that χ(A(0))=t2+μt+λ has a root αJ(R) and a root βU(R).

Write χ(A(x))=t2+μ(x)t+λ(x) where μ(x)= i=0μixi,λ(x)= i=0λixiR[[x]] and μ0=μ,λ0=λ. Let b0=α. It is easy to verify that μ0=α+βU(R). Hence, 2b0+μ0U(R). Choose

b1=(2b0+μ0)1(λ1μ1b0),b2=(2b0+μ0)1(λ2μ1b1μ2b0b12),

Then y= i=0bixiR[[x]] is a root of χ(A(x)). In addition, yJ(R[[x]]) as b0J(R). Since y2+μ(x)y+λ(x)=0, we have χ(A(x))=(ty)(t+y)+μ(ty)=(ty)(t+y+μ). Set z=yμ. Then zU(R[[x]]) as μU(R[[x]]). Therefore χ(A(x)) has a root in J(R[[x]]) and a root in U(R[[x]]). According to Theorem 2.4, A(x)M2(R[[x]]) is strongly rad-clean, as asserted.

Corollary 3.3. Let R be a commutative local ring. Then the following are equivalent:

• (1) Every AM2(R) with invertible trace is strongly rad-clean.

• (2) Every A(x1,,xn)M2(R[[x1,,xn]]) with invertible trace is strongly rad-clean.

Proof. (1)(2) Let A(x1,,xn)M2(R[[x1,,xn]]) with invertible trace. Then trA(0,,0)U(R). By hypothesis, A(0,,0)M2(R) is strongly rad-clean. In light of Theorem 2.4, A(x1,,xn)M2(R[[x1,,xn]]) is strongly rad-clean.

(2)(1) is obvious.

### References

1. D. D. Anderson and V. P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra, 30(7)(2002), 3327-3336.
2. G. Borooah, A. J. Diesl and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra, 212(1)(2008), 281-296.
3. S. Breaz, Matrices over finite fields as sums of periodic and nilpotent elements, Linear Algebra Appl., 555(2018), 92-97.
4. H. Chen. Rings Related to Stable Range Conditions. Series in Algebra 11. NJ: World Scientific, Hackensack; 2011.
5. J. Cui and J. Chen, When is a 2×2 matrix ring over a commutative local ring quasipolar?, Comm. Algebra, 39(9)(2011), 3212-3221.
6. P. Danchev, Certain properties of square matrices over fields with applications to rings, Rev. Colombiana Mat., 54(2)(2020), 109-116.
7. P. Danchev, E. Garcia and M. G. Lozano, Decompositions of matrices into potent and square-zero matrices, Internat. J. Algebra Comput., 32(2)(2022), 251-263.
8. A. J. Diesl and T. J. Dorsey, Strongly clean matrices over arbitrary rings, J. Algebra, 399(2014), 854-869.
9. Y. Li, Strongly clean matrix rings over local rings, J. Algebra, 312(1)(2007), 397-404.
10. J. Šter, On expressing matrices over 2 as the sum of an idempotent and a nilpotent, Linear Algebra Appl., 544(10)(2018), 339-349.
11. G. Tang, Y. Zhou and H. Su, Matrices over a commutative ring as sums of three idempotents or three involutions, Linear Multilinear Algebra, 67(2)(2019), 267-277.