CC BY
60
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2019. Т. 27. С. 28-35
УДК 512.552.13
MSG 16U99; 16Е50; 13В99
DOI https://doi.org/10.26516/1997-7670.2019.27.28
Left-Right Cleanness and Nil Cleanness in Unital Rings
P. V. Danchev
Institute of Mathematics and Informatics of Bulgarian Academy of Sciences, Sofia, Bulgaria
Abstract. We introduce the notions of left and right cleanness and nil cleanness in rings showing their close relationships with the classical concepts of cleanness and nil cleanness. Specifically, it is proved that strongly clean rings are both L-clean and R-clean as well as strongly nil clean rings are both L-nil clean and R-nil clean. These two assertions somewhat strengthen well-known results due to Nicholson (Comm. Algebra, 1999) and Diesl (J. Algebra, 2013). Moreover, it is shown that L-nil cleanness (respectively, R-nil cleanness) is preserved modulo nil Jacobson radical as well as that this is still true for L-cleanness (respectively, R-cleanness), provided the Jacobson radical is nil.
Keywords: clean rings, nil clean rings, L-clean rings, R-clean rings, L-nil clean rings, R-nil clean rings.
Everywhere in the text of the current short paper, all rings P under consideration are assumed to be associative, containing the identity element 1, which differs from the zero element 0 of P, and all subrings are unital (i.e., containing the same identity as that of the former ring). Our standard notations and the terminology are mainly in agreement with [6]. Concretely, U(P) denotes the set of all units in P, Id(P) the set of all idempotents in P, Nil(P) the set of all nilpotents in P and J(P) the Jacobson radical of P. For the defined in the sequel left, (resp., right) cleanness and nil cleanness notions, we shall use for our convenience the letters "L" and "R", respectively
1. Introduction and Background
Referring to [7] a ring P is said to be clean if, for each r£P, there exist e € Id(P) and u € U(P) such that r = e + u. If, in addition, eu = ue, P is called strongly clean. The class of clean rings contains all unit-regular rings (see [2]), whereas the class of strongly clean rings encompasses all strongly 7r-regular rings (see [7]); note that strongly 7r-regular rings are themselves unit-regular whenever they are (von Neumann) regular. Recently, it was constructed a concrete example in [9] of a unit-regular ring which is not strongly clean.
On the other hand, mimicking [5], a ring P is said to be nil clean if, for every rGP, there exist e € Id(P) and q € Nil(P) such that r = e + q. If, in addition, eq = qe, P is called strongly nil clean. The class of (strongly) nil clean rings is properly contained in the class of (strongly) clean rings as well as strongly nil clean rings are themselves strongly 7r-regular (cf. [5]).
Our brief article is motivated by one more fundamental property of cleanness and nil cleanness related to one-sided ideals in rings. The newly obtained properties somewhat increase the description of the so-difficult general structure of clean and nil clean rings.
2. L-clean and R-clean Rings
We start here with some new variations of cleanness.
Definition 1. Let P be a ring. We will say that P is L-clean if, for any x € P, there are e € (1 — x)P n Id(P) and u € U(P) such that x = e + u. Analogically, if e € P(1 —x)C\ Id{P), P is said to be R-clean.
By definition, both L-clean and R-clean rings are necessarily clean. Moreover, the critical elements 0 and 1 have the following trivial representations as both L-clean and R-clean elements: 0 = 0 + 1 with 0 = 1.0 = 0.1 € IP n PI; 1 = 0 + 1 with 0 = 0.1 = 1.0 € OP n P0.
What we next may comment, is that Definition 1 is tantamount to the following equivalent reformulations:
For any y e P, setting x = 1 — y, it must be that e € yP fl Id(P) and that y = (1 — e) + v, where v = —u € U(P). By replacing 1 — e with /, we detect that it amounts to 1 — / € yP fl Id(P) with y = f + v.
By analogy, e € Py fl Id(P) and y = (1 — e) + v, where v = —u € U(P). By substituting 1—e with /, we receive that it amounts to 1—/ € PyDld(P) with y = f + v. With the aid of these equivalencies, idempotents and units can be presented in the sense of L-clean and R-clean elements as follows: e = (1 — e) + (2e — 1) with e = e.l = l.e € ePn Pe; u = 0 + u = (1 - 1) +u with 1 = uu~l = «"'a £ uP fl Pu, as required.
What can be observed at once as a valuable example of such rings is that strongly regular rings are both L-clean and R-clean (for a common
generalization, see Corollary 1 stated below). Indeed, if P is a strongly regular ring, then one can write x = eu = ue for some e € Id(P) and и € U(P) such that x = (1 - e) + (ue - 1 + e) € Id(P) + U(P) with e = xu~l = u~lx € xP П Px, as wanted.
Nevertheless, the following implication, which extends the last observation, could be useful.
Proposition 1. Strongly clean rings are both L-clean and R-clean.
Proof. For any element x of such a ring P, one writes that 1-х = e+u with eu = ue for some idempotent e and unit и from P. Thus x = (1 — e) — и and xe = ue, so that one can verify that e = u~lxe = xu~le € xP, because x and и do commute, as asked for.
The case of R-cleanness can be handled in a similar way □
As an immediate consequence, we derive the following statement. However, we will give a new more conceptual confirmation of its validity
Corollary 1. Strongly n-regular rings are both L-clean and R-clean.
Proof. According to [7, Proposition 1], one deduces for such a ring P that xn = eu = ue for some n € N, e € Id(P) and и € U(P), whenever x is an arbitrary element from P. Thus e = жга-и-1 € xP and one can concludes as demonstrated in the already cited Proposition 1 from [7] that there is w = x — 1 + еб(7 (P) such that x = (1 — e) + w, as desired.
Paralleling, R-cleanness follows as well. □
In accordance with [10, Corollary 4], for all ngN, the upper triangular n x n matrix ring Tn(K) over the ring К = £ Q : n is odd}, where Q is the field of all rationals, is strongly clean but neither strongly 7r-regular nor local.
The following folklore fact, pertaining to a little more specific lifting of idempotents modulo a nil ideal, is a key instrument for our next theorem: If К is a ring with a nil ideal I, d € К and d+I € Id(K/I), then d+I = e+I for some e € Id(K) П dK with de = ed.
We are now in a position to prove the following statement.
Theorem 1. Suppose that P is a ring with nil J(P). Then P is L-clean (resp., R-clean) if, and only if, P/J(P) is L-clean (resp., R-clean).
Proof. "Necessity." Setting that P = P/J(P) and, for all x £ P, that x = x + J(P), we obtain x € P. If we, furthermore, write x = 1 —e + u for e € Id(P) П xP and и € U(P), then one derives that x = 1 — e + й. Likewise, ё = e + J(P) = xy + J(P) = (x + J(P))(y + J(P)) = xye xP, for some у € P, as asserted.
"Sufficiency." Given x € P, we have x € P and write that x = [1—a]+b with a € xP, for some idempotent a and some unit b in P. Therefore, one writes that x + J(P) = [1 - a + J(P)] + [b + J(P)] = (1 - a + b) + J(P), where it is clear that b € i/(P) since 1 + J(P) < U(P). Also, it is not too hard to check that U(P) + J(P) = U(P) holds. Consequently, one deduces that x = 1 — a + u for some u € Nil(P). Besides, a + J(P) = (a; + J(P))(c + J(P)) = xc + J(P) is an idempotent and hence, as J(P) is by assumption nil, we utilize the posted above folklore fact, to find that a + J(P) = e + J(P) for some idempotent e € P having the property that e € (xc)P ç xP. Thus a € e + J(P) and, finally, x € 1 - e + U(P), as claimed.
In parallel to above arguments, R-cleanness also follows. □
Remark 1. In connection with Corollary 1, it seems that unit-regular rings are eventually hardly L-clean (resp., R-clean). In fact, it was showed in [2] that a ring K is unit-regular if, and only if, V a; € if : x = e + u for some e € Id(K) and u € U(K) such that xR n eR = {0}. If, additionally, they were L-clean (or R-clean, respectively), it must be that e € (1 — x)K. With the exchange property at hand, 1 — e € xK. All of these inclusions maybe should interpret some contradiction. However, a concrete example of a unit-regular ring which is not L-clean (resp., R-clean) is not presently constructed yet. In case this can be made, in virtue of Proposition 1 that construction will substantially refine the aforementioned corresponding example from [9] of a unit-regular non strongly clean ring (compare with Problem 2 quoted below).
3. L-nil clean and R-nil clean Rings
We begin here with some new variations of nil cleanness.
Definition 2. Let P be a ring. We will say that P is L-nil clean if, for any x € P, there are e € xP n Id(P) and q € Nil(P) such that x = e + q. Analogously, if e € Px n Id{P), P is said to be R-nil clean.
By definition, both L-nil clean and R-nil clean rings are of necessity nil clean. The truthfulness of the converse implication is in question yet (compare with Problem 1 posed below).
Obvious examples of both L-nil clean and R-nil clean rings are the boolean rings B, the indecomposable ring Z4, the upper triangular 2x2 matrix ring ¥2(22). A reason for this is that their elements are idempo-tents, nilpotents and unipotents only, and these special elements have the following L-nil clean and R-nil clean presentations in an arbitrary ring P:
• For any e € Id(P), we write:
e = e + 0 and e = e.l € eP, as required.
—e = e + (—2e), where the second term is a nilpotent as so is 2, and e = (—e).(—1) € (—e)P, as required.
• For any q € Nil(P), we write:
q = 0 + q and 0 = q.O € qP, as required.
1 + q = 1 + q and 1 = (1 + q).( 1 + q)~l € (1 + q)P, as required.
Even nil clean units can be presented thus: и = e + q with e = и — q = u( 1 — u~lq) € uP and e = и — q = (1 — qu~l)u € Pu, as required.
What we can further see is that L-nil clean rings (resp., R-nil clean rings) are always L-clean (resp., R-clean). In fact, writing as above that x = e + q with e € Id(P) П xP and q € Nil(P), one observes that x = (l-e) + (2e-l + q) £ Id(P) + U(P) as 2 e Nil(P) and so 2e + q£ Nil(P). Similarly can be processed in the case of R-nil cleanness.
The following folklore fact, already used above in Theorem 1, concerning a little more special lifting of idempotents modulo a nil ideal, is crucial for our next theorem: If К is a ring with a nil ideal I, d € К and d + I € Id(K/I), then d + I = e + I for some e € Id(K) П dK with de = ed.
We are now ready to prove the following assertion.
Theorem 2. Suppose P is a ring. Then P is L-nil clean (resp., R-nil clean) if, and only if, P/J(P) is L-nil clean (resp., R-nil clean) and J(P) is nil.
Proof. "Necessity." That the ideal J(P) is nil follows at once applying [5], because as just already noticed L-nil clean and R-nil clean rings are both nil clean. Putting now that P = P/J(P) and, for all x € P, that x = x + J(P), we have x € P. If we, furthermore, write x = e + q for e € Id(P) П xP and q € Nil(P), then one detects that x = e + q. Likewise, ё = e + J(P) = xy + J(P) = (x + J(P))(y + J(P)) =xye xP, for some у € P, as expected.
We can process by the same token and for R-nil cleanness.
"Sufficiency." Given x € P, we have x € P and write that x = a + b with a € xP, for some idempotent a and some nilpotent b in P. Therefore, one writes that x+J(P) = [a+J(P)] + [b+J(P)\ = (a+b)+J(P), where it is clear that b € Nil(P) since J(P) is nil. Also, it is not too hard to check that b+J(P) = {b+j | j € J(P)} С Nil(P) holds. Consequently, one infers that ж = a + q for some q € Nil(P). Besides, a + J{P) = (x + J(P))(c + J(P)) = xc + J(P) is an idempotent and hence, utilizing the listed above folklore fact, we get that a+J(P) = e+J(P) for some idempotent e € P having the property that e € (xc)P С xP. Thus a € e+J(P) and, finally, x € e+J(P), as promised.
We can process in the same manner and for R-nil cleanness. □
As an immediate consequence, we yield:
Corollary 2. Strongly nil clean rings are both L-nil clean and R-nil clean.
Proof. It was established in [4] (see [8] too) that a ring P is strongly nil clean if, and only if, P/J(P) is a boolean ring and J(P) is a nil ideal. As aforementioned, boolean rings are always L-nil clean and R-nil clean, so that we can apply Theorem 2 to arrive at our claim. □
Remark 2. It is worthwhile noticing that the upper triangular matrix 2x2 ring ¥2(22) is always strongly nil clean. In view of the stated above element-wise presentations accomplished with results from [1], a non strongly nil clean example of both L-nil clean and R-nil clean rings is the full matrix 2x2 ring M2(^2) whose elements consist of only nil clean units, idempotents and nilpotents. However, ¡^2(^-2) is strongly clean.
Moreover, it follows directly from Theorem 2 that 11^2(^4) is both L-nil clean and R-nil clean. Indeed, one sees that Z4/J(Z4) = Z2 and so M2(Z2) ^ M2(Z4/J(Z4)) = M2(Z4)/M2(J(Z4)) = M2(Z4)/J(M2(Z4)), where J(Z4) = {0,2} is nil whence so does J(M2(Z4)) = M2(J(Z4)). As already noted above, ¡^2(^-2) is both L-nil clean and R-nil clean, so that we are set.
On the other vein, Corollary 2 could be proved more transparently as follows: Writing in the presence of above notations that x = e + q with eq = qe, and thus xe = ex, we quickly obtain by expanding x — e with k, where qk = 0, k € N, that is (x — e)k = 0, that e € xP as well as that e € Px, as needed.
We end our work with the following two questions of some interest and importance:
Problem 1. Is it true that nil clean rings are L-nil clean or, respectively, R-nil clean?
Certainly, for any nil clean ring P with x € P one writes that x = e + q, where e € P is an idempotent and q G P is a nilpotent, but does there exist a record x = f +1 for an idempotent / and a nilpotent t such that / € xP (resp., / € Px) is not too elementary.
Problem 2. Does it follow that unit-regular rings are L-clean (resp., R-clean)? Generally, does there exist a clean ring which is neither L-clean nor R-clean?
It is worth noticing that, owing to Proposition 1, such a ring has to be not strongly clean. In this observation in mind, let we consider the subring K = € Q : n is odd} of the field of rational numbers Q and also the 2x2 full matrix ring V\2(K) over K. It was shown in [10, Example 1] that V\2(K) is a semiperfect (and hence clean by [3, Theorem 9]) ring which is
not strongly clean. Let now we test Y\2(K) for L-cleanness and R-cleanness, expecting that this will definitely be false. To that goal, let us consider the
non-invertible matrix ^ , which must be presented as the sum of a nontrivial idempotent and a unit as the matrix ^ ^ = - ^ ^ is nonunit as well. Simple calculations show that the non-trivial idempotents are of the form 1 b_ , where a,b,c € K with be = a — a2. Thus
8 6\ _ fa b \ f 8 — a 6 — 6 3 7 J ~ \ c I — a J I 3 — c 6 + a
On the other vein, a few more not too straightforward computations should lead to a contradiction that
but at this stage this is impossible to be verified.
4. Concluding Discussion
We shall state here a brief summary of all established above results, namely:
(i) All strongly clean (and, in particular, strongly 7r-regular) rings are both L-clean and R-clean.
(ii) All strongly nil clean rings are both L-nil clean and R-nil clean.
(iii) Let ii be a ring with nil J(K). Then K is L-clean (respectively, R-clean) K/J(K) is L-clean (respectively, R-clean).
(iv) Let ii be a ring. Then K is L-nil clean (respectively, R-nil clean) K/J(K) is L-nil clean (respectively, R-nil clean) and J(K) is nil.
References
1. Breaz S., Calugareanu G., Danchev P., Micu T. Nil-clean matrix rings. Lin. Algebra & Appl., 2013, vol. 439, pp. 3115-3119. https://doi.Org/10.1016/j.laa.2013.08.027
2. Camillo V.P., Khurana D. A characterization of unit regular rings. Commun. Algebra, 2001, vol. 29, pp. 2293-2295. https://doi.org/10.1081/AGB-100002185
3. Camillo V.P., Yu H.P. Exchange rings, units and idempotents. Commun. Algebra, 1994, vol. 22, pp. 4737-4749. https://doi.org/10.1080/00927879408825098
4. Danchev P.V., Lam T.Y. Rings with unipotent units. Publ. Math. Debrecen, 2016, vol. 88, pp. 449-466. https://doi.org/10.5486/PMD.2016.7405
5. Diesl A.J. Nil clean rings. J. Algebra, 2013, vol. 383, pp. 197-211. https://doi.Org/10.1016/j.jalgebra.2013.02.020
6. Lam T.Y. A First Course in Noncommutative Rings. Second Edition, Graduate Texts in Math., 2001, vol. 131, Springer-Verlag, Berlin-Heidelberg-New York. https://doi.org/10.1007/978-l-4419-8616-0
7. Nicholson W. K. Strongly clean rings and Fitting's lemma. Commun. Algebra, 1999, vol. 27, pp. 3583-3592. https://doi.org/10.1007/978-l-4419-8616-0
8. Kosan T., Wang Z., Zhou Y. Nil-clean and strongly nil-clean rings. J. Pure and Appl. Algebra, 2016, vol. 220, pp. 633-646.
9. Nielsen P.P., Ster J. Connections between unit-regularity, regularity, cleanness and strong cleanness of elements and rings. Trans. Amer. Math. Soc., 2018, vol. 370, pp. 1759-1782. https://doi.org/10.1090/tran/7080
10. Wang Z., Chen J. On two open problems about strongly clean rings. Bull. Austral. Math. Soc., 2004, vol. 70, pp. 279-282. https://doi.org/10.1017/S0004972700034493
Peter Vassilev Danchev, Ph.D. (Mathematics), Professor, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113, Acad. G. Bonchev str., bl. 8, Sofia, Bulgaria (e-mail: danchevOmath. bas. bg; pvdanchevOyahoo. com)
Received 08.12.18
L-чистота, R-чистота и нильпотентная чистота унитарных колец
П. В. Данчев
Институт математики и информатики Болгарской академии наук, София, Болгария
Аннотация. В статье определены и исследованы два новых понятия, а именно классы L-чистых и R-чистых колец, а также классы L-нильпотентно и R-нильпотен-тно чистых колец. Полученные результаты уточняют некоторые классические результаты Николсона (Comm. Algebra, 1999) и Дизеля (J. Algebra, 2013).
Ключевые слова: чистые кольца, нильпотентно чистые кольца, L-чистота кольца, R-чистота кольца.
Петр Васильевич Данчев, кандидат физико-математических наук, Институт математики и информатики Болгарской академии наук, Болгария, 1113, София, ул. Акад. Г. Бончева, бл. 8 (e-mail: danchev@math.bas.bg; pvdanchev@yahoo.com)
Поступила в редакцию 08.12.18
