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When is every linear transformation a sum of two commuting invertible ones?
Linear Algebra and its Applications 439 (2013) 3615–3619
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Linear Algebra and its Applications www.elsevier.com/locate/laa
When is every linear transformation a sum of two commuting invertible ones? Gaohua Tang a,∗ , Yiqiang Zhou b a
School of Mathematical Science, Guangxi Teachers Education University, Nanning 530023, PR China Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld A1C 5S7, Canada b
a r t i c l e
i n f o
Article history: Received 21 June 2013 Accepted 23 September 2013 Available online 9 October 2013 Submitted by P. Semrl MSC: 15A04 15A23 16U60 Keywords: Linear transformation 2-Sum property Strong 2-sum property Units Strongly π -regular endomorphism Semisimple module
a b s t r a c t A well-known result of Wolfson [7] and Zelinsky [8] says that every linear transformation of a vector space V over a division ring D is a sum of two invertible linear transformations except when dim( V ) = 1 and D = F2 . Indeed, many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. The goal of this note is to prove that every linear transformation of a vector space V over a division ring D is a sum of two commuting invertible ones if and only if | D | 3 and dim( V ) < ∞. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Wolfson [7] in 1953 and Zelinsky [8] in 1954, independently, proved that every linear transformation of a vector space V over a division ring D is a sum of two invertible linear transformations except when dim( V ) = 1 and D = F2 . This result promoted considerable interest in the rings generated by their units (see, for example, [2,3,5,6] and the references there). In this note, we observe that many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. This property has a connection with the notion of a strongly
*
Corresponding author. E-mail addresses: [email protected] (G. Tang), [email protected] (Y. Zhou).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.09.038
3616
G. Tang, Y. Zhou / Linear Algebra and its Applications 439 (2013) 3615–3619
clean ring by Nicholson [4, Proposition 2], where a ring is called strongly clean if each element of the ring is a sum of a unit and an idempotent that commute. The goal here is to prove that every linear transformation of a vector space V over a division ring D is a sum of two commuting invertible ones if and only if | D | 3 and dim( V ) < ∞. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms. Throughout, rings are associative with identity, and modules are unitary right modules. For a module M over a ring R, End R ( M ) stands for the endomorphism ring of M. Homomorphisms of modules are written on the left of their arguments. The set of units of a ring R is denoted by U ( R ). We write Mn ( R ) for the n × n matrix ring over R. The cardinal of a set X is denoted | X |. 2. The results An element of a ring is said to satisfy the 2-sum property if it is a sum of two units of the ring and the ring is said to satisfy the 2-sum property if each of its elements satisfies the 2-sum property. By the terminology of Vámos [6], an element (respectively, a ring) with the 2-sum property is called a 2-good element (respectively, 2-good ring). Definition 1. An element a of a ring R is said to satisfy the strong 2-sum property if a is a sum of two units that commute with each other. A ring R is said to have the strong 2-sum property if every element of R satisfies the strong 2-sum property. Let Q ( R ) be the set of quasi-regular elements of R. That is, Q ( R ) = {q ∈ R: q + p + qp = q + p + pq = 0 for some p ∈ R }. Note that q ∈ Q ( R ) ⇔ 1 + q ∈ U ( R ). Lemma 2 below can be easily proved. Lemma 2. The following hold for a ring R: (1) Every element of Q ( R ) satisfies the strong 2-sum property. (2) Every central element of R with the 2-sum property satisfies the strong 2-sum property. (3) Let σ : R → S be an isomorphism of rings. If a ∈ R satisfies the strong 2-sum property, then σ (a) ∈ S satisfies the strong 2-sum property. (4) Let R = α ∈Λ R α be a direct product of rings. Then R has the strong 2-sum property iff R α has the strong 2-sum property for all α ∈ Λ. Lemma 3. Let D be a division ring. Then D has the strong 2-sum property iff | D | 3. Proof. The implication in one direction is clear. Suppose | D | 3. The zero element 0 is certainly a sum of two commuting units. Moreover, 1 − u = 0 for some 0 = u ∈ D, so 1 = (1 − u ) + u is a sum of two commuting units. For 0 = a ∈ D with a = 1, a = 1 + (a − 1) is a sum of two commuting units. 2 An element a in a ring R is strongly π -regular if an ∈ an+1 R ∩ Ran+1 for some n 1. A ring is called strongly π -regular if each element is strongly π -regular. Strongly π -regular rings include one-sided perfect rings. In particular, one-sided Artinian rings are strongly π -regular. In the next lemma, the equivalence (1) ⇔ (2) is due to Armendariz, Fisher and Snider [1, Proposition 2.31], and the equivalence (1) ⇔ (3) can be found in Nicholson [4]. Lemma 4. The following are equivalent for f ∈ End R ( M ): (1) f is strongly π -regular. (2) M = Im( f n ) ⊕ Ker( f n ) for some n 1. (3) There is a decomposition M = A ⊕ B, where A and B are f -invariant, f | A is a unit in End R ( A ) and f | B is nilpotent in End R ( B ).
G. Tang, Y. Zhou / Linear Algebra and its Applications 439 (2013) 3615–3619
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We are now ready to prove our main result. Theorem 5. Let V be a vector space over a division ring D. Then End D ( V ) has the strong 2-sum property iff | D | 3 and dim( V ) < ∞. Proof. ( ⇒). First we prove | D | 3. Assume on the contrary that D = F2 . Write V = v D ⊕ W , where 0 = v ∈ V and W is a subspace of V . Let f ∈ End D ( V ) be given by f ( v ) = v and f ( W ) = 0. We show that f does not have the strong 2-sum property. To see this, suppose that f = α + β where α , β are units of End D ( V ) and α β = β α . From f α = α f , it follows that α ( v D ) = α ( f ( v D )) = f (α ( v D )) ⊆ v D. Similarly, β( v D ) ⊆ v D. As D = F2 and α , β are invertible, α ( v ) = v and β( v ) = v. So v = f ( v ) = (α + β)( v ) = α ( v ) + β( v ) = v + v = 0. This is a contradiction. Hence we have proved | D | 3. Next we prove dim( V ) < ∞. Assume on the contrary ∞ that dim( V ) = ∞. Let { v i : i = 1, 2, . . .} be a set of linearly independent vectors of V and let U = i =1 v i D. Write V = U ⊕ W . Define σ ∈ End D ( V ) by σ ( v i ) = v i +1 for i = 1, 2, . . . and σ (x) = x for x ∈ W . We have the following: Claim. If h ∈ End D ( V ) with hσ = σ h, then h( W ) ⊆ W . Proof of Claim. For any x ∈ W , write h(x) = v 1 d1 + v 2 d2 + · · · + v n dn + y where y ∈ W and di ∈ D for i = 1, 2, . . . . Then h(x) = hσ (x) = σ h(x) = σ ( v 1 d1 + v 2 d2 + · · · + v n dn + y ) = v 2 d1 + v 3 d2 + · · · + v n+1 dn + y, so d1 = d2 = · · · = dn = 0. Thus h(x) = y ∈ W . Now we show that σ does not have the strong 2-sum property. Assume that there exist α , β ∈ U (End D ( V )) such that σ = α + β and α β = β α . As α (α −1 ( v 1 )) = α −1 (α ( v 1 )) = v 1 ∈ / W , α(v 1) ∈ /W / W by the claim. Write and α −1 ( v 1 ) ∈
α ( v 1 ) = v 1a1 + · · · + v n an + x and α −1 ( v 1 ) = v 1 b1 + · · · + v m bm + x , where x, x ∈ W and ai , b j ∈ D for i = 1, . . . , n and j = 1, . . . , m with an = 0 and bm = 0. Then, as ασ = σ α ,
v 1 = αα −1 ( v 1 )
= α v 1 b1 + v 2 b2 + · · · + v m bm + x
= α ( v 1 )b1 + α ( v 2 )b2 + · · · + α ( v m )bm + α x = α ( v 1 )b1 + α σ ( v 1 ) b2 + · · · + α σ m−1 ( v 1 ) bm + α x = α ( v 1 )b1 + σ α ( v 1 ) b2 + · · · + σ m−1 α ( v 1 ) bm + α x = v 1a1 b1 + v 2 (a2 b1 + a1 b2 ) + · · · + v n+m−1an bm + xb1 + σ (x)b2 + · · · + σ m−1 (x)bm + α x . As x, x ∈ W , xb1 + σ (x)b2 + · · · + σ m−1 (x)bm + α (x ) ∈ W by the claim. As an bm = 0, it follows that n + m − 1 = 1. So n = m = 1. Hence α ( v 1 ) = v 1 a1 + x. A similar argument shows that β( v 1 ) = v 1 c 1 + y where c 1 ∈ D and y ∈ W . Now from σ = α + β , it follows that v 2 = σ ( v 1 ) = α ( v 1 ) + β( v 1 ) = v 1 (a1 + c 1 ) + (x + y ), a contradiction.
(⇐ ). We prove the claim by induction on dimension of the vector space V . If dim( V ) = 1, the claim is true by Lemma 3. Assume that dim( V ) = n 2 and that the claim holds true for any vector space over D of dimension less than n. Let f ∈ End D ( V ). We next show that f has the strong 2-sum property. This is certainly true if f is nilpotent by Lemma 2(1). So we can assume that f is not nilpotent in End D ( V ). Case 1: f is not a unit in End D ( V ). As f ∈ End D ( V ) is strongly π -regular, by Lemma 4 there is a decomposition V = A ⊕ B where A and B are f -invariant, f | A is a unit in End D ( A ) and f | B is nilpotent in End D ( B ). We can see that A = 0 and B = 0, so dim( A D ) < n and dim( B D ) < n. By induction hypothesis, f | A = α1 + β1 and f | B = α2 + β2 where α1 , β1 are commuting units in End D ( A ) and α2 , β2
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are commuting units in End D ( B ). Define α , β ∈ End D ( V ) by setting α | A = α1 , β| B = β2 . Then f = α + β is a sum of two commuting units in End D ( V ).
α | B = α2 and β| A = β1 ,
Case 2: f is a unit in End D ( V ), but 1 − f is not nilpotent in End D ( V ). If 1 − f is a unit of End D ( V ), then f = 1 + ( f − 1) is a sum of two commuting units. So we can assume that 1 − f is not a unit in End D ( V ). As 1 − f ∈ End D ( V ) is strongly π -regular, there is a decomposition V = A ⊕ B where A and B are (1 − f )-invariant, (1 − f )| A is a unit in End D ( A ) and (1 − f )| B is nilpotent in End D ( B ). Thus, A = 0 and B = 0, so dim( A D ) < n and dim( B D ) < n. Moreover, A , B being (1 − f )-invariant implies that A , B are f -invariant. By induction hypothesis, f | A = α1 + β1 and f | B = α2 + β2 where α1 , β1 are commuting units in End D ( A ) and α2 , β2 are commuting units in End D ( B ). Define α , β ∈ End D ( V ) by setting α | A = α1 , α | B = α2 and β| A = β1 , β| B = β2 . Then f = α + β is a sum of two commuting units in End D ( V ). Case 3: 1 − f is nilpotent in End D ( V ). Let g = 1 − f . If g = 0, then f = 1. Since | D | 3, f has the 2-sum property by [8], so f has the strong 2-sum property by Lemma 2(2). Hence we can assume g = 0. Let m be a positive integer minimal with respect to g m = 0. Then m 2. As g m−1 = 0, there exists v ∈ V such that g m−1 ( v ) = 0. Then v , g ( v ), . . . , g m−1 ( v ) are linearly independent over D, and A := v D + g ( v ) D + · · · + g m−1 ( v ) D is a g-invariant subspace of V D . Let ρ : V D → D D be a D-linear mapping such that ρ ( g m−1 ( v )) = 0. Let B = ker(ρ ) ∩ ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 ). Then B is a g-invariant subspace of V D . One easily sees that A ∩ B = 0. Since dim(ker(ρ )) = n − 1, ker(ρ ) is a maximal subspace of V . As g m−1 ( V ) ⊆ ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 ) and g m−1 ( V ) ker(ρ ), we have ker(ρ ) + (ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 )) = V , and hence n = dim( V ) = dim(ker(ρ )) + dim(ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 )) − dim( B ). That is,
dim( B ) + 1 = dim ker(ρ ◦ g ) ∩ · · · ∩ ker
ρ ◦ g m −1 .
As dim(ker(ρ ◦ g )) = n − 1 and g m−2 ( V ) ⊆ ker(ρ ◦ g 2 ) ∩ · · · ∩ ker(ρ ◦ g m−1 ) and g m−2 ( V ) ker(ρ ◦ g ), we also have ker(ρ ◦ g ) + (ker(ρ ◦ g 2 ) ∩ · · · ∩ ker(ρ ◦ g m−1 )) = V . As above, one deduces
dim( B ) + 2 = dim
ρ ◦ g 2 ∩ · · · ∩ ker ρ ◦ g m−1 .
Continuing the process, we obtain dim( B ) + (m − 1) = dim(ker(ρ ◦ g m−1 )). As dim(ker(ρ ◦ g m−1 )) = n − 1, it follows that dim( B D ) = n − m. Therefore, V = A ⊕ B. If B = 0, then dim( A D ) < n and dim( B D ) < n. Moreover, A , B are f -invariant. As argued in Case 1 or Case 2, f is a sum of two commuting units in End D ( V ). So we can assume that B = 0, i.e., m = n. With respect to the basis v , g ( v ), . . . , g n−1 ( v ) of V , f = 1 − g can be identified with the matrix
⎛ 1 −1 0 1
··· 0 0 ··· 0 0
0 0 0 0
··· 1 −1 ··· 0 1
⎞
⎜. . ⎟ . . . ⎟ A=⎜ ⎝ .. .. . . .. .. ⎠ in Mn ( D ). By Lemma 2(3), to finish the proof, it suffices to show that A ∈ Mn ( D ) satisfies the strong 2-sum property. Since | D | 3, take 0 = a ∈ D with a = 1. Then
⎛ 1 − a −1 ⎞ ··· 0 1−a ⎜ 0 ··· 0⎟ ⎜ . .. ⎟+⎜ . A=⎜ . . . . . ⎝ .. .. . . .. ⎠ ⎜ . ⎝ 0 0 0 0 ··· a ⎛
a 0 ⎜0 a
0
0
is a sum of two commuting units in Mn ( D ).
··· ··· .. .
0 0
0 0
⎞
⎟ .. .. ⎟ ⎟ . . ⎟ · · · 1 − a −1 ⎠ ··· 0 1−a
2
With slight efforts, we can extend Theorem 5 to semisimple modules. A module M R is called semisimple if M R is a direct sum of simple R-modules, or equivalently if every submodule of M R is a direct summand. First let us assume that M R is a homogeneous semisimple module, and let N be a simple submodule. Then M ∼ = N ( I ) , where I is an index set, and End R ( M ) ∼ = CFM I ( D ), the ring of I × I -column finite matrices over D, where D = End R ( N ) is a division ring. As CFM I ( D ) ∼ = End D ( D ( I ) ),
G. Tang, Y. Zhou / Linear Algebra and its Applications 439 (2013) 3615–3619
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by Theorem 5 we see that End R ( M ) has the strong 2-sum property iff I is finite and | D | 3. Since N R is simple, there is a non-trivial automorphism of N R iff there exist x, y ∈ N with x = y such that x⊥ = y ⊥ , where x⊥ and y ⊥ are the annihilators of x and y in R, respectively (in this case, xr → yr, ∀r ∈ R, gives a non-trivial automorphism of N R ). Now if M R is a semisimplemodule, then ∼ M= Λ M α is the direct sum of homogeneous components of M, so End R ( M ) = Λ End R ( M α ). Thus, we deduce that End R ( M ) has the strong 2-sum property iff End R ( M α ) has the strong 2-sum property for all α ∈ Λ. In summary, we obtain the following. Corollary 6. Let M be a semisimple module over a ring R. Then End R ( M ) has the strong 2-sum property iff every homogeneous component of M is finitely generated and every simple submodule of M contains two distinct elements with the same annihilator in R. Acknowledgements The research of the first author was supported by the National Natural Science Foundation of China (11161006), the Guangxi Natural Science Foundation (2011GXNSFA018139) and the Guangxi new century 1000 talent project, and the second author by a Discovery Grant from NSERC of Canada. References [1] E.P. Armendariz, J.W. Fisher, R.L. Snider, On injective and surjective endomorphisms of finitely generated modules, Comm. Algebra 6 (1978) 659–672. [2] F. Barroero, C. Frei, R.F. Tichy, Additive unit representations in global fields – a survey, Publ. Math. Debrecen 79 (3–4) (2011) 291–307. [3] C. Meehan, Sums of automorphisms of free modules and completely decomposable groups, J. Algebra 299 (2006) 467–479. [4] W.K. Nicholson, Strongly clean rings and fitting’s lemma, Comm. Algebra 27 (8) (1999) 3583–3592. [5] A.K. Srivastava, A survey of rings generated by units, Ann. Fac. Sci. Toulouse Math. 19 (2010) 203–213. [6] P. Vámos, 2-good rings, Q. J. Math. 56 (3) (2005) 417–430. [7] K.G. Wolfson, An ideal-theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953) 358–386. [8] D. Zelinsky, Every linear transformation is a sum of nonsingular ones, Proc. Amer. Math. Soc. 5 (1954) 627–630.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
When is every linear transformation a sum of two commuting invertible ones? Gaohua Tang a,∗ , Yiqiang Zhou b a
School of Mathematical Science, Guangxi Teachers Education University, Nanning 530023, PR China Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld A1C 5S7, Canada b
a r t i c l e
i n f o
Article history: Received 21 June 2013 Accepted 23 September 2013 Available online 9 October 2013 Submitted by P. Semrl MSC: 15A04 15A23 16U60 Keywords: Linear transformation 2-Sum property Strong 2-sum property Units Strongly π -regular endomorphism Semisimple module
a b s t r a c t A well-known result of Wolfson [7] and Zelinsky [8] says that every linear transformation of a vector space V over a division ring D is a sum of two invertible linear transformations except when dim( V ) = 1 and D = F2 . Indeed, many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. The goal of this note is to prove that every linear transformation of a vector space V over a division ring D is a sum of two commuting invertible ones if and only if | D | 3 and dim( V ) < ∞. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Wolfson [7] in 1953 and Zelinsky [8] in 1954, independently, proved that every linear transformation of a vector space V over a division ring D is a sum of two invertible linear transformations except when dim( V ) = 1 and D = F2 . This result promoted considerable interest in the rings generated by their units (see, for example, [2,3,5,6] and the references there). In this note, we observe that many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. This property has a connection with the notion of a strongly
*
Corresponding author. E-mail addresses: [email protected] (G. Tang), [email protected] (Y. Zhou).
0024-3795/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.09.038
3616
G. Tang, Y. Zhou / Linear Algebra and its Applications 439 (2013) 3615–3619
clean ring by Nicholson [4, Proposition 2], where a ring is called strongly clean if each element of the ring is a sum of a unit and an idempotent that commute. The goal here is to prove that every linear transformation of a vector space V over a division ring D is a sum of two commuting invertible ones if and only if | D | 3 and dim( V ) < ∞. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms. Throughout, rings are associative with identity, and modules are unitary right modules. For a module M over a ring R, End R ( M ) stands for the endomorphism ring of M. Homomorphisms of modules are written on the left of their arguments. The set of units of a ring R is denoted by U ( R ). We write Mn ( R ) for the n × n matrix ring over R. The cardinal of a set X is denoted | X |. 2. The results An element of a ring is said to satisfy the 2-sum property if it is a sum of two units of the ring and the ring is said to satisfy the 2-sum property if each of its elements satisfies the 2-sum property. By the terminology of Vámos [6], an element (respectively, a ring) with the 2-sum property is called a 2-good element (respectively, 2-good ring). Definition 1. An element a of a ring R is said to satisfy the strong 2-sum property if a is a sum of two units that commute with each other. A ring R is said to have the strong 2-sum property if every element of R satisfies the strong 2-sum property. Let Q ( R ) be the set of quasi-regular elements of R. That is, Q ( R ) = {q ∈ R: q + p + qp = q + p + pq = 0 for some p ∈ R }. Note that q ∈ Q ( R ) ⇔ 1 + q ∈ U ( R ). Lemma 2 below can be easily proved. Lemma 2. The following hold for a ring R: (1) Every element of Q ( R ) satisfies the strong 2-sum property. (2) Every central element of R with the 2-sum property satisfies the strong 2-sum property. (3) Let σ : R → S be an isomorphism of rings. If a ∈ R satisfies the strong 2-sum property, then σ (a) ∈ S satisfies the strong 2-sum property. (4) Let R = α ∈Λ R α be a direct product of rings. Then R has the strong 2-sum property iff R α has the strong 2-sum property for all α ∈ Λ. Lemma 3. Let D be a division ring. Then D has the strong 2-sum property iff | D | 3. Proof. The implication in one direction is clear. Suppose | D | 3. The zero element 0 is certainly a sum of two commuting units. Moreover, 1 − u = 0 for some 0 = u ∈ D, so 1 = (1 − u ) + u is a sum of two commuting units. For 0 = a ∈ D with a = 1, a = 1 + (a − 1) is a sum of two commuting units. 2 An element a in a ring R is strongly π -regular if an ∈ an+1 R ∩ Ran+1 for some n 1. A ring is called strongly π -regular if each element is strongly π -regular. Strongly π -regular rings include one-sided perfect rings. In particular, one-sided Artinian rings are strongly π -regular. In the next lemma, the equivalence (1) ⇔ (2) is due to Armendariz, Fisher and Snider [1, Proposition 2.31], and the equivalence (1) ⇔ (3) can be found in Nicholson [4]. Lemma 4. The following are equivalent for f ∈ End R ( M ): (1) f is strongly π -regular. (2) M = Im( f n ) ⊕ Ker( f n ) for some n 1. (3) There is a decomposition M = A ⊕ B, where A and B are f -invariant, f | A is a unit in End R ( A ) and f | B is nilpotent in End R ( B ).
G. Tang, Y. Zhou / Linear Algebra and its Applications 439 (2013) 3615–3619
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We are now ready to prove our main result. Theorem 5. Let V be a vector space over a division ring D. Then End D ( V ) has the strong 2-sum property iff | D | 3 and dim( V ) < ∞. Proof. ( ⇒). First we prove | D | 3. Assume on the contrary that D = F2 . Write V = v D ⊕ W , where 0 = v ∈ V and W is a subspace of V . Let f ∈ End D ( V ) be given by f ( v ) = v and f ( W ) = 0. We show that f does not have the strong 2-sum property. To see this, suppose that f = α + β where α , β are units of End D ( V ) and α β = β α . From f α = α f , it follows that α ( v D ) = α ( f ( v D )) = f (α ( v D )) ⊆ v D. Similarly, β( v D ) ⊆ v D. As D = F2 and α , β are invertible, α ( v ) = v and β( v ) = v. So v = f ( v ) = (α + β)( v ) = α ( v ) + β( v ) = v + v = 0. This is a contradiction. Hence we have proved | D | 3. Next we prove dim( V ) < ∞. Assume on the contrary ∞ that dim( V ) = ∞. Let { v i : i = 1, 2, . . .} be a set of linearly independent vectors of V and let U = i =1 v i D. Write V = U ⊕ W . Define σ ∈ End D ( V ) by σ ( v i ) = v i +1 for i = 1, 2, . . . and σ (x) = x for x ∈ W . We have the following: Claim. If h ∈ End D ( V ) with hσ = σ h, then h( W ) ⊆ W . Proof of Claim. For any x ∈ W , write h(x) = v 1 d1 + v 2 d2 + · · · + v n dn + y where y ∈ W and di ∈ D for i = 1, 2, . . . . Then h(x) = hσ (x) = σ h(x) = σ ( v 1 d1 + v 2 d2 + · · · + v n dn + y ) = v 2 d1 + v 3 d2 + · · · + v n+1 dn + y, so d1 = d2 = · · · = dn = 0. Thus h(x) = y ∈ W . Now we show that σ does not have the strong 2-sum property. Assume that there exist α , β ∈ U (End D ( V )) such that σ = α + β and α β = β α . As α (α −1 ( v 1 )) = α −1 (α ( v 1 )) = v 1 ∈ / W , α(v 1) ∈ /W / W by the claim. Write and α −1 ( v 1 ) ∈
α ( v 1 ) = v 1a1 + · · · + v n an + x and α −1 ( v 1 ) = v 1 b1 + · · · + v m bm + x , where x, x ∈ W and ai , b j ∈ D for i = 1, . . . , n and j = 1, . . . , m with an = 0 and bm = 0. Then, as ασ = σ α ,
v 1 = αα −1 ( v 1 )
= α v 1 b1 + v 2 b2 + · · · + v m bm + x
= α ( v 1 )b1 + α ( v 2 )b2 + · · · + α ( v m )bm + α x = α ( v 1 )b1 + α σ ( v 1 ) b2 + · · · + α σ m−1 ( v 1 ) bm + α x = α ( v 1 )b1 + σ α ( v 1 ) b2 + · · · + σ m−1 α ( v 1 ) bm + α x = v 1a1 b1 + v 2 (a2 b1 + a1 b2 ) + · · · + v n+m−1an bm + xb1 + σ (x)b2 + · · · + σ m−1 (x)bm + α x . As x, x ∈ W , xb1 + σ (x)b2 + · · · + σ m−1 (x)bm + α (x ) ∈ W by the claim. As an bm = 0, it follows that n + m − 1 = 1. So n = m = 1. Hence α ( v 1 ) = v 1 a1 + x. A similar argument shows that β( v 1 ) = v 1 c 1 + y where c 1 ∈ D and y ∈ W . Now from σ = α + β , it follows that v 2 = σ ( v 1 ) = α ( v 1 ) + β( v 1 ) = v 1 (a1 + c 1 ) + (x + y ), a contradiction.
(⇐ ). We prove the claim by induction on dimension of the vector space V . If dim( V ) = 1, the claim is true by Lemma 3. Assume that dim( V ) = n 2 and that the claim holds true for any vector space over D of dimension less than n. Let f ∈ End D ( V ). We next show that f has the strong 2-sum property. This is certainly true if f is nilpotent by Lemma 2(1). So we can assume that f is not nilpotent in End D ( V ). Case 1: f is not a unit in End D ( V ). As f ∈ End D ( V ) is strongly π -regular, by Lemma 4 there is a decomposition V = A ⊕ B where A and B are f -invariant, f | A is a unit in End D ( A ) and f | B is nilpotent in End D ( B ). We can see that A = 0 and B = 0, so dim( A D ) < n and dim( B D ) < n. By induction hypothesis, f | A = α1 + β1 and f | B = α2 + β2 where α1 , β1 are commuting units in End D ( A ) and α2 , β2
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are commuting units in End D ( B ). Define α , β ∈ End D ( V ) by setting α | A = α1 , β| B = β2 . Then f = α + β is a sum of two commuting units in End D ( V ).
α | B = α2 and β| A = β1 ,
Case 2: f is a unit in End D ( V ), but 1 − f is not nilpotent in End D ( V ). If 1 − f is a unit of End D ( V ), then f = 1 + ( f − 1) is a sum of two commuting units. So we can assume that 1 − f is not a unit in End D ( V ). As 1 − f ∈ End D ( V ) is strongly π -regular, there is a decomposition V = A ⊕ B where A and B are (1 − f )-invariant, (1 − f )| A is a unit in End D ( A ) and (1 − f )| B is nilpotent in End D ( B ). Thus, A = 0 and B = 0, so dim( A D ) < n and dim( B D ) < n. Moreover, A , B being (1 − f )-invariant implies that A , B are f -invariant. By induction hypothesis, f | A = α1 + β1 and f | B = α2 + β2 where α1 , β1 are commuting units in End D ( A ) and α2 , β2 are commuting units in End D ( B ). Define α , β ∈ End D ( V ) by setting α | A = α1 , α | B = α2 and β| A = β1 , β| B = β2 . Then f = α + β is a sum of two commuting units in End D ( V ). Case 3: 1 − f is nilpotent in End D ( V ). Let g = 1 − f . If g = 0, then f = 1. Since | D | 3, f has the 2-sum property by [8], so f has the strong 2-sum property by Lemma 2(2). Hence we can assume g = 0. Let m be a positive integer minimal with respect to g m = 0. Then m 2. As g m−1 = 0, there exists v ∈ V such that g m−1 ( v ) = 0. Then v , g ( v ), . . . , g m−1 ( v ) are linearly independent over D, and A := v D + g ( v ) D + · · · + g m−1 ( v ) D is a g-invariant subspace of V D . Let ρ : V D → D D be a D-linear mapping such that ρ ( g m−1 ( v )) = 0. Let B = ker(ρ ) ∩ ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 ). Then B is a g-invariant subspace of V D . One easily sees that A ∩ B = 0. Since dim(ker(ρ )) = n − 1, ker(ρ ) is a maximal subspace of V . As g m−1 ( V ) ⊆ ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 ) and g m−1 ( V ) ker(ρ ), we have ker(ρ ) + (ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 )) = V , and hence n = dim( V ) = dim(ker(ρ )) + dim(ker(ρ ◦ g ) ∩ · · · ∩ ker(ρ ◦ g m−1 )) − dim( B ). That is,
dim( B ) + 1 = dim ker(ρ ◦ g ) ∩ · · · ∩ ker
ρ ◦ g m −1 .
As dim(ker(ρ ◦ g )) = n − 1 and g m−2 ( V ) ⊆ ker(ρ ◦ g 2 ) ∩ · · · ∩ ker(ρ ◦ g m−1 ) and g m−2 ( V ) ker(ρ ◦ g ), we also have ker(ρ ◦ g ) + (ker(ρ ◦ g 2 ) ∩ · · · ∩ ker(ρ ◦ g m−1 )) = V . As above, one deduces
dim( B ) + 2 = dim
ρ ◦ g 2 ∩ · · · ∩ ker ρ ◦ g m−1 .
Continuing the process, we obtain dim( B ) + (m − 1) = dim(ker(ρ ◦ g m−1 )). As dim(ker(ρ ◦ g m−1 )) = n − 1, it follows that dim( B D ) = n − m. Therefore, V = A ⊕ B. If B = 0, then dim( A D ) < n and dim( B D ) < n. Moreover, A , B are f -invariant. As argued in Case 1 or Case 2, f is a sum of two commuting units in End D ( V ). So we can assume that B = 0, i.e., m = n. With respect to the basis v , g ( v ), . . . , g n−1 ( v ) of V , f = 1 − g can be identified with the matrix
⎛ 1 −1 0 1
··· 0 0 ··· 0 0
0 0 0 0
··· 1 −1 ··· 0 1
⎞
⎜. . ⎟ . . . ⎟ A=⎜ ⎝ .. .. . . .. .. ⎠ in Mn ( D ). By Lemma 2(3), to finish the proof, it suffices to show that A ∈ Mn ( D ) satisfies the strong 2-sum property. Since | D | 3, take 0 = a ∈ D with a = 1. Then
⎛ 1 − a −1 ⎞ ··· 0 1−a ⎜ 0 ··· 0⎟ ⎜ . .. ⎟+⎜ . A=⎜ . . . . . ⎝ .. .. . . .. ⎠ ⎜ . ⎝ 0 0 0 0 ··· a ⎛
a 0 ⎜0 a
0
0
is a sum of two commuting units in Mn ( D ).
··· ··· .. .
0 0
0 0
⎞
⎟ .. .. ⎟ ⎟ . . ⎟ · · · 1 − a −1 ⎠ ··· 0 1−a
2
With slight efforts, we can extend Theorem 5 to semisimple modules. A module M R is called semisimple if M R is a direct sum of simple R-modules, or equivalently if every submodule of M R is a direct summand. First let us assume that M R is a homogeneous semisimple module, and let N be a simple submodule. Then M ∼ = N ( I ) , where I is an index set, and End R ( M ) ∼ = CFM I ( D ), the ring of I × I -column finite matrices over D, where D = End R ( N ) is a division ring. As CFM I ( D ) ∼ = End D ( D ( I ) ),
G. Tang, Y. Zhou / Linear Algebra and its Applications 439 (2013) 3615–3619
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by Theorem 5 we see that End R ( M ) has the strong 2-sum property iff I is finite and | D | 3. Since N R is simple, there is a non-trivial automorphism of N R iff there exist x, y ∈ N with x = y such that x⊥ = y ⊥ , where x⊥ and y ⊥ are the annihilators of x and y in R, respectively (in this case, xr → yr, ∀r ∈ R, gives a non-trivial automorphism of N R ). Now if M R is a semisimplemodule, then ∼ M= Λ M α is the direct sum of homogeneous components of M, so End R ( M ) = Λ End R ( M α ). Thus, we deduce that End R ( M ) has the strong 2-sum property iff End R ( M α ) has the strong 2-sum property for all α ∈ Λ. In summary, we obtain the following. Corollary 6. Let M be a semisimple module over a ring R. Then End R ( M ) has the strong 2-sum property iff every homogeneous component of M is finitely generated and every simple submodule of M contains two distinct elements with the same annihilator in R. Acknowledgements The research of the first author was supported by the National Natural Science Foundation of China (11161006), the Guangxi Natural Science Foundation (2011GXNSFA018139) and the Guangxi new century 1000 talent project, and the second author by a Discovery Grant from NSERC of Canada. References [1] E.P. Armendariz, J.W. Fisher, R.L. Snider, On injective and surjective endomorphisms of finitely generated modules, Comm. Algebra 6 (1978) 659–672. [2] F. Barroero, C. Frei, R.F. Tichy, Additive unit representations in global fields – a survey, Publ. Math. Debrecen 79 (3–4) (2011) 291–307. [3] C. Meehan, Sums of automorphisms of free modules and completely decomposable groups, J. Algebra 299 (2006) 467–479. [4] W.K. Nicholson, Strongly clean rings and fitting’s lemma, Comm. Algebra 27 (8) (1999) 3583–3592. [5] A.K. Srivastava, A survey of rings generated by units, Ann. Fac. Sci. Toulouse Math. 19 (2010) 203–213. [6] P. Vámos, 2-good rings, Q. J. Math. 56 (3) (2005) 417–430. [7] K.G. Wolfson, An ideal-theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953) 358–386. [8] D. Zelinsky, Every linear transformation is a sum of nonsingular ones, Proc. Amer. Math. Soc. 5 (1954) 627–630.