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АЛГЕБРО-ЛОГИЧЕСКИЕ МЕТОДЫ В ИНФОРМАТИКЕ И ИСКУССТВЕННЫЙ ИНТЕЛЛЕКТ
ALGEBRAIC AND LOGICAL METHODS IN COMPUTER SCIENCE AND ARTIFICIAL INTELLIGENCE
Серия «Математика»
2019. Т. 29. С. 3-9
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
ИЗВЕСТИЯ
Иркутского государственного ■университета
УДК 512.552.13
MSG 16U99; 16Е50; 13В99
DOI https://doi.Org/10.26516/1997-7670.2019.29.3
A Note on Commutative Nil-Clean Corners in Unital Rings
P. V. Danchev
Institute of Mathematics and Informatics of Bulgarian Academy of Sciences, Sofia, Bulgaria
Abstract. We shall prove that if R is a ring with a family of orthogonal idempotents {ег}"= i having sum 1 such that each corner subring etRet is commutative nil-clean, then R is too nil-clean, by showing that this assertion is actually equivalent to the statement established by Breaz-Cälugäreanu-Danchev-Micu in Lin. Algebra & Appl. (2013) that if R is a commutative nil-clean ring, then the full matrix ring is also nil-clean
for any size n. Likewise, the present proof somewhat supplies our recent result in Bull. Iran. Math. Soc. (2018) concerning strongly nil-clean corner rings as well as it gives a new strategy for further developments of the investigated theme.
Keywords: nil-clean rings, nilpotents, idempotents, corners
1. Introduction and Background
Everywhere in the text of this short paper, our rings R into consideration are assumed to be associative, containing the identity element 1 which, in general, differs from the zero element 0 of R, and all proper subrings are unital (i.e., containing the same identity as that of the former ring) with the exception of the corners eRe having the identity e, where e is an arbitrary
idempotent of R. Our terminology and notations are at all standard being mainly in agreement with [11]. As usual, J{R) stands for the Jacobson radical of R, and Mra(_R) for the full n x n matrix ring over R whenever n € IN.
First of all, let us recall two more notions, namely that a ring R is called boolean whenever r2 = r for every r € R, and is called strongly regular whenever, for every r € R, there exists an element a € R such that r = r2a. Evidently, boolean rings are always strongly regular, while the converse is false.
On the other side, a ring R is said to be nil-clean in [9] (see, e.g., [8] as well) if, for every element r € R, there exist a nilpotent q and an idempotent e, both depending on r, with the property r = q + e; such an element r is also called nil-clean. In [1] it was shown that the matrix ring Mn{F) over a field F is nil-clean if, and only if, F = h2. This was slightly extended to the matrix ring over a division ring in [10]. In particular, if R is a commutative nil-clean ring, then Mn(R) remains nil-clean as well (see [2], too) as well as if R is a strongly regular ring, then Mra(_R) is nil-clean if, and only if, R is a boolean ring.
The aim that we pursue here is to give a brief note on nil-cleanness of corner rings. We shall demonstrate that the facts presented above are deducible without any matrix at hand; in fact, we shall use a few new tricks in terms of corners of an arbitrary ring (see [3], [4], [5] and [6] as well). As for the converse, it was asked in [9] of whether or not the nil-cleanness of R will also imply nil-cleanness of the corner subring eRe for any idempotent e of R, that question seems to be rather difficult and so it leaves unanswered yet.
2. The Main Result
Before proving up our main achievement, we need two technical claims as follows:
Proposition 1. Every finitely generated subring of a commutative nil-clean ring is finite.
Proof. Let R be a commutative nil-clean ring and let S be its subring generated by the elements a\, ■ ■ ■ ,an € R. Write ец = e^ + qi for each index i, where e^ is an idempotent and qi is a nilpotent, and consider the subring So of R generated by all such a and qi. Choosing now an integer к such that qf = 0 for every index i, we observe that So as an additive group is generated by a finite set
Iе 11 e22 ''' etQiQi2 ''' Qln I 0 < h < 1; 0 < h < к - 1}.
Since by [9] the char(E) is finite, it follows immediately that So must be a finitely generated torsion group and, therefore, it has to be finite. Hence S C So is finite, too. □
Remark 1. The above assertion could also be derived in the following more conceptual manner: It follows easily from [9] that a commutative ring R is nil-clean exactly when J(R) is nil and the quotient R/J(R) is boolean (see [7] as well). As any subring of a boolean ring is again boolean, it now plainly follows that a subring of a commutative nil-clean ring is also nil-clean. Thus being simultaneously finitely generated, we routinely obtain its finiteness.
Nevertheless, the illustrated above proof gives some further strategies to be developed.
The following technicality is pivotal for obtaining and proving our main result.
Lemma 1. Let R be a ring with a family of pair-wise orthogonal idem-potents ei,...,en € R with YH=iei = 1- V each corner ring eiRei is commutative nil-clean, then, for every a € R, the subring of R generated by the set {ei,..., en, a} is finite.
Proof We shall verify the statement by the usage of an induction on the number of idempotents n. In doing that, for n = 1, the conclusion follows appealing to Proposition 1. So, let us assume that n > 2 and fix an element a € R. For each i = 1,..., n, let Ri be the subring of the corner (1—ei)R(l — e^, generated by the set {e1} e2,..., e^-i, ei+1,..., en, (1 - ei)a(l - e^j. Notice that Ri has to be finite by the induction hypothesis. Let us now Si be the subring of eiRei, generated by the union {ei, eiaei} UeiaRiaei. Since Si is obviously a finitely generated subring of a commutative nil-clean ring, it is finite by Proposition 1. Note also that the element e^ete^ae^ ■ ■ ■ aeikaei lies in Si for every 1 <i\,...,ik <n.
Now, let T be the subring of R, generated by the set {e\,... ,en,a}. Then T, as an additive group, is generated by the set
T = {e^ae^ • • • aeik | k > 1,1 < ij < n, 1 < j < k}.
We assert that this set is finite. Indeed, take x = e^ae^ • • • aeik € T. Choose j > 1 such that ij = i\ and j is maximal with this property Consequently, x = ras, where r = e^ae^ • • • ae^. € S^ and
S — ' ' ' ^^ik ^ -^il
provided j / k, or x € Sh provided j = k. Thus T Q [U^SilUlU^SiaRi] is really finite, and hence T is finitely generated as an additive group. Note that, applying [9], the element 2e^ is a nilpotent in the corner eiRei for
each i, so that 2 is a nilpotent in R. Hence T is a finitely generated torsion group, whence finite, as expected. □
We now have all the ingredients necessary to proceed by proving our chief statement.
Theorem 1. Suppose that R is a ring with a family of pair-wise orthogonal idempotents e\,..., en € R with Ym=i = 1- V eac^ corner r^n9 e^-Re^ is commutative nil-clean, then R is also nil-clean.
Proof. Assuming that the ring R satisfies the assumptions of the text, we take an arbitrary element a € R. Let us now S be the subring of R, generated by the finite system {ei,..., en, a}. We will prove now that S is necessarily nil-clean. In fact, with Lemma 1 at hand, the ring S is finite, whence by the well-known structure characterization of artinian rings accomplished with the classical Wedderburn's theorem (see cf. [11] for instance), one may write that
S/J(S) ^ Mrai(Fi) x • • • x Mnr(Fr) = T,
where all rij > 1 and all Fj are finite fields (j e[l,r],r£fi). It is not too difficult to observe that, under this isomorphism, the complete orthogonal set of idempotents ei,..., en is mapped into the complete orthogonal set of idempotents Щ,..., e^ € T, and the corresponding corner rings Ti = ё^ГЩ are commutative nil-clean. Note also that
Ti = Mrai^(Fi) x • • • x Mnr .(Fr),
where it is pretty obvious that 0 < rij^ < щ. The commutativity of the ring Ti forces Hjti < 1 for any index j, whereas, in accordance with [1, Theorem 3], the nil-cleanness of Ti implies that Fj = whenever rij^ = 1. Hence, if Fj Щ Zj2 for some j, we would have rij^ = 0 for all i, thus contradicting the completeness of the family {ei,..., en}. Therefore, Fj = for all j. Furthermore, once again using [1, Theorem 3], we shall obtain that T is nil-clean. Since S is finite, one easily sees that J{S) is nil, whence with the aid of [9] it follows that S is nil-clean, indeed, as claimed.
That the ring R is also nil-clean follows rather elementary as the element a is already shown to be nil-clean, concluding the proof. □
As an immediate consequence, one deduces the following assertion which is actually the chief result in [1] (compare with [2] too). Besides, this manifestly shows that the above theorem and the next corollary are tantamount being deducible one of other.
Corollary 1. If R is a commutative nil-clean ring, then Mn{R) is nil-clean.
Proof. It is principally known that there is a system of n pair-wise orthogonal idempotents of Mn(R), say f\,...,fn, with sum equal to the identity matrix En such that the isomorphisms R = /iMra(_R)/i = • • • = fnM-n{R)fn hold. Henceforth, the previous Theorem 1 successfully works to get the wanted claim. □
3. Concluding Discussion
As final comments, it is worthwhile noticing that we somewhat have obtain some advantage on the problem raised by Diesl in [9] of whether the full matrix ring of any size over a nil-clean ring is again nil-clean. As noticed above, this was partially settled in [1] when the former ring R is commutative nil-clean, that is, R/J(R) is boolean and J(R) is nil.
The further progress in the object to solve completely the aforementioned Diesl's problem seems to be not too quick by taking into account the very complicated structure of non-commutative nil-clean rings (see, e.g., [7]).
We end our work with the following query consisting of two questions of interest:
Problem 1. What can be said for the ring R from Theorem 1, provided that its corners eiRei are non-commutative nil-clean rings? Is this ring R still nil-clean?
References
1. Breaz S., Calugareanu G., Danchev P., Micu T. Nil-clean matrix rings, Lin. Algebra & Appl, 2013, vol. 439, no. 10, pp. 3115-3119. https://doi.Org/10.1016/j.laa.2013.08.027
2. Danchev P. V. Strongly nil-clean corner rings, Bull. Iran. Math. Soc., 2017, vol. 43, no. 5, pp. 1333-1339.
3. Danchev P.V. Semi-boolean corner rings, Internat. Math. Forum, 2017, vol. 12, no. 16, pp. 795-802. https://doi.org/10.12988/imf.2017.7655
4. Danchev P.V. On corner subrings of unital rings, Internat. J. Contemp. Math. Sci., 2018, vol. 13, no. 2, pp. 59-62. https://doi.org/10.12988/ijcms.2018.812
5. Danchev P.V. Corners of invo-clean unital rings, Pure Math. Sci., 2018, vol. 7, no. 1, pp. 27-31. https://doi.org/10.12988/pms.2018.877
6. Danchev P.V. Feebly nil-clean unital rings, Proc. Jangjeon Math. Soc., 2018, vol. 21, no. 1, pp. 155-165.
7. Danchev P.V., Lam T.Y. Rings with unipotent units, Publ. Math. Debrecen, 2016, vol. 88, no. 3-4, pp. 449-466. https://doi.org/10.5486/PMD.2016.7405
8. Danchev P.V., McGovern W.Wm. Commutative weakly nil clean unital rings, J. Algebra, 2015, vol. 425, no. 5, pp. 410-422. https://doi.Org/10.1016/j.jalgebra.2014.12.003
9. Diesl A.J. Nil clean rings, J. Algebra, 2013, vol. 383, no. 11, pp. 197-211. https://doi.Org/10.1016/j.jalgebra.2013.02.020
10. Kosan M.T., Lee T.K., Zhou Y. When is every matrix over a division ring a sum of an idempotent and a nilpotent?, Lin. Algebra & Appl, 2014, vol. 450, no. 11, pp. 7-12. https://doi.Org/10.1016/j.laa.2014.02.047
11. Lam T.Y. A First Course in Noncommutative Rings, Second Edition, Graduate Texts in Math., 2001, vol. 131, Springer-Verlag, Berlin-Heidelberg-New York.
Peter Vassilev Danchev, Ph.D. (Mathematics), Professor, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., bl. 8, 1113, Sofia, Bulgaria (e-mail: danchevOmath. bas. bg; pvdanchevOyahoo. com)
Received 01.08.19
О коммутативных ниль-чистых угловых подкольцах в унитарных кольцах
П. В. Данчев
Институт математики и информатики Болгарской академии наук, София, Болгария
Аннотация. Мы доказали, что если R — кольцо с семейством ортогональных идемпотентов {e¿}"=1, имеющее сумму 1, такую, что каждое угловое подкольдо e¿iíe¿ коммутативно ниль-чисто, тогда R также ниль-чисто, показывая, что это утверждение фактически эквивалентно утверждению, установленному Breaz S., Cälugäreanu G., Danchev P., Micu T. в "Lin. Algebra & Appl." (2013), что если R — коммутативное ниль-чистое кольцо, то полное матричное кольцо М„(Д) также ниль — чисто для любого размера те. Настоящее доказательство в некоторой степени уточняет наш недавний результат, опубликованный в журнале "Bull. Iran. Math. Soc. "(2018), касающийся сильно ниль-чистых угловых колец, а также дает новую стратегию для дальнейшего развития исследуемой темы.
Ключевые слова: ниль-чистые кольца, нильпотенты, идемпотенты, угловые подкольца.
Список литературы
1. Breaz S., Calugareanu G., Danchev P., Micu T. Nil-clean Matrix Rings // Lin. Algebra & Appl. 2013. Vol. 439, N 10. P. 3115-3119. https://doi.Org/10.1016/j.laa.2013.08.027
2. Danchev P. V. Strongly nil-clean corner rings // Bull. Iran. Math. Soc. 2017. Vol. 43, N 5. P. 1333-1339.
3. Danchev P. V. Semi-boolean corner rings // Internat. Math. Forum. 2017. Vol. 12, N 16. P. 795-802. https://doi.org/10.12988/imf.2017.7655
4. Danchev P. V. On corner subrings of unital rings // Internat. J. Contemp. Math. Sci. 2018. Vol. 13, N 2. P. 59-62. https://doi.org/10.12988/ijcms.2018.812
5. Danchev P. V. Corners of invo-clean unital rings // Pure Math. Sci. 2018. Vol. 7, N 1. P. 27-31. https://doi.org/10.12988/pms.2018.877
6. Danchev P. V. Feebly nil-clean unital rings // Proc. Jangjeon Math. Soc., 2018, Vol. 21, N 1. P. 155-165.
7. Danchev P. V., Lam T. Y. Rings with Unipotent Units // Publ. Math. Debrecen. 2016. Vol. 88, N 3-4. P. 449-466. https://doi.org/10.5486/PMD.2016.7405
8. Danchev P. V., McGovern W. Wm. Commutative weakly nil clean unital rings // J. Algebra. 2015. Vol. 425, N 5. P. 410-422. https://doi.Org/10.1016/j.jalgebra.2014.12.003
9. Diesl A. J. Nil Clean Rings // J. Algebra. 2013. Vol. 383, N 11. P. 197-211. https://doi.Org/10.1016/j.jalgebra.2013.02.020
10. Ko§an M. Т., Lee Т. K., Zhou Y. When is every matrix over a division ring a sum of an idempotent and a nilpotent? // Lin. Algebra & Appl. 2014. Vol. 450, N 11. P. 7-12. https://doi.Org/10.1016/j.laa.2014.02.047
11. Lam T. Y. A First Course in Noncommutative Rings. Second Edition. Graduate Texts in Math. 2001. Vol. 131. Springer-Verlag, Berlin-Heidelberg-New York.
Петр Васильевич Данчев, доктор физико-математических наук, Институт математики и информатики Болгарской академии наук, Болгария, 1113, г. София, ул. Акад. Г. Бончев, бл. 8 (e-mail: danchev@math.bas.bg; pvdanchev@yahoo.com)
Поступила в редакцию 01.08.19
