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Endomorphisms of free modules as sums of four quadratic endomorphisms

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Simion Breaz at Babeş-Bolyai University
Simion Breaz
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Abstract

We will prove that every endomorphism of a free right R-module of infinite rank can be decomposed as a sum of four endomorphisms which satisfy some fixed polynomial identities.
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ENDOMORPHISMS OF FREE MODULES AS SUMS OF FOUR
QUADRATIC ENDOMORPHISMS
SIMION BREAZ
Abstract. We will prove that every endomorphism of a free right R-module
of infinite rank can be decomposed as a sum of four endomorphisms which
satisfy some fixed polynomial identities.
The study of the problem of decomposing an endomorphism of an infinitely
generated free module as a sum of endomorphisms which satisfy various identities
is a natural extension of the research areas on various decompositions of matrices
(we refer to some classical papers as [4], [5], and to some new results proved in
[1], [2], [3] or [10]) or bounded operators on Hilbert spaces (see [7], [12], [13]). In
the last years there has been significant progress in this area. In order to describe
this, let us recall that if p1, . . . , p`are polynomials over the center of a ring R, an
endomorphism αof an R-module is a (p1, . . . , p`)-sum if it has a decomposition
P`
k=1 αksuch that pk(αk) = 0 for all k= 1, `. If all pkare of degree 2 we say that
αis a sum of quadratic endomorphisms, [8].
The main results included in the above mentioned progress are the following:
de Seguins Pazzis developed a technique which uses stratifications of modules over
polynomial rings in order to prove that in the case of infinite-dimensional vec-
tor spaces over fields for every four split quadratic polynomials p1, . . . , p4, every
endomorphism is a (p1, p2, p3, p4)-sum, [8, Theorem 2]; Shitov proved, by using
combinatorial techniques applied to infinite directed graphs, that for every polyno-
mials pk=X2−mkX,k= 1,4, with mkin the center of R, every endomorphism
of a non-finitely generated free module is a (p1, p2, p3, p4)-sum, [11, Remark 11].
Moreover, de Seguins Pazzis provided in [9] an extensive study of endomorphisms
of infinite-dimensional vector spaces that are (p1, p2, p3)-sums. From [9, Theorem 3]
it follows that the above mentioned results are optimal in the sense that there exists
a field K, three split quadratic polynomials p1, p2, p3over K, and an endomorphism
αof an infinite-dimensional K-vector space such that αis not a (p1, p2, p3)-sum.
In this note we present a simple and transparent proof for a generalization of
[8, Theorem 2] and [11, Remark 11] for free right modules of infinite rank. This is
based on ideas used by Harris in [5], respectively by Pearcy and Topping in [7]. We
are able to use these ideas since Mesyan proved in [6] that every endomorphism ϕ
of an infinitely generated free right R-module Fis a commutator, i.e. there exist
x, y ∈End(F) such that ϕ=xy −yx.
MSC2010: 15A24
Keywords: free module, endomorphism, commutator, (p1,...,p`)-sum.
1
2 SIMION BREAZ
Theorem 1. Let Rbe a ring. For every k= 1,4we consider two elements mk
and nkfrom the center of R, and we denote pk= (X−mk)(X−nk). Then every
endomorphism of a free right R-module Fof infinite rank is a (p1, p2, p3, p4)-sum.
Proof. Since Fis of infinite rank, we have F∼
=F⊕F, hence we can view every
endomorphism of Fas a matrix α=a b
c d with a, b, c, d ∈End(F).
By [6, Proposition 2.14] there exist two endomorphisms x, y ∈End(F) such that
a+d−P4
k=1(mk+nk) = xy −yx. Applying the same decomposition techniques
as those used in [5] and [7], it is easy to see that there exist z, u, v ∈End(F) such
that
α=xy +m1x
(n1−m1)y−yxy n1−yx +n2−z n2−m2−z
z m2+z
+n3u
0m3+m40
v n4.
This equality represents a decomposition of αas a (p1, p2, p3, p4)-sum, and the proof
is complete.
Acknowledgements. I would like to thank the referee for the very detailed re-
port I received. Referee’s suggestions helped me substantially improve the result I
present here.
References
[1] Abyzov, A.N., Mukhametgaliev, I.I.: On some matrix analogs of the little Fermat theorem,
Math. Notes 101 (2017), 187–192.
[2] Breaz, S., C˘alug˘areanu, Gr.: Sums of nilpotent matrices, Linear Multilinear Algebra 65
(2017), 67–78.
[3] Breaz S., Modoi G.C.: Nil-clean companion matrices. Linear Algebra Appl. 489 (2016),
50–60.
[4] Fillmore, P.A.: On Similarity and the Diagonal of a Matrix. Amer. Math. Monthly 76 (1969),
167–169.
[5] Harris B.: Commutators in division rings. Proc. Amer. Math. Soc. 4(1958), 628–630.
[6] Mesyan K.: Commutator rings, Bull. Austral. Math. Soc. 74 (2006), 279–288.
[7] Pearcy, C., Topping, D.: Sums of small numbers of idempotents, Michigan Math. J. 14
(1967), 453–465.
[8] de Seguins Pazzis, C.: Sums of quadratic endomorphisms of an infinite-dimensional vector
space, to appear in Proc. Edinburgh Math. Soc.; preprint arXiv:1601.00296 [math.RA].
[9] de Seguins Pazzis, C.: Sums of three quadratic endomorphisms of an infinite-dimensional
vector space, Acta Sci. Math. (Szeged) 83 (2017), 83–111.
[10] de Seguins Pazzis, C.: A note on sums of three square-zero matrices, Linear Multilinear
Algebra 65 (2017), 787–805.
[11] Shitov, Y.: Sums of square-zero endomorphisms of a free module, Linear Algebra Appl. 507
(2016), 191–197.
[12] Stampfli, J.G.: Sums of pro jections, Duke Math. J. 31 (1964), 455–461.
[13] Wang J.-H., Wu P.Y.: Sums of square-zero operators. Studia Math. 99 (2) (1991), 115–127.
Department of Mathematics, Babes¸ -Bolyai University, 1 Kog˘
alniceanu street, Cluj-
Napoca, Romania
E-mail address:bodo@math.ubbcluj.ro
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