We prove that a ring R is exchange 2-UU if, and only if, J(R) is nil and R/J(R)≅B×C, where B is a Boolean ring and C is a ring with C ⊆ Πμ ℤ₃ for some ordinal μ. We thus somewhat improve on a result due to Abdolyousefi-Chen (J. Algebra Appl., 2018) by showing that it is a simple consequence of already well-known results of Danchev-Lam (Publ. Math. Debrecen, 2016) and Danchev (Commun. Korean Math. Soc., 2017)

Danchev Peter V

University of Toyama Repository

Toyama Math. J.Vol. 39(2017), 1-7On exchange -UU unital ringsPeter V. DanchevAbstract. We prove that a ring R is exchange 2-UU if, and onlyif, J(R) is nil and R=J(R) = B  C, where B is a Boolean ring andC is a ring with C  Q Z3 for some ordinal . We thus somewhatimprove on a result due to Abdolyouse-Chen (J. Algebra Appl., 2018)by showing that it is a simple consequence of already well-knownresults of Danchev-Lam (Publ. Math. Debrecen, 2016) and Danchev(Commun. Korean Math. Soc., 2017).1. Introduction and BackgroundEverywhere in the text of the current article, all our rings are assumed tobe associative, containing the identity element 1 which diers from the zeroelement 0. Our terminology and notations are mainly in agreement withthe stated in [7]. For instance, for an arbitrary ring R, U(R) will alwaysdenote the unit group with n-th power Un(R) = fun j u 2 U(R)g, wheren 2 N, J(R) the Jacobson radical, and Nil(R) the set of all nilpotents.Recall also that a ring R is said to be tripotent provided that the equalityx3 = x holds for all x 2 R.We also need some other fundamentals as follows:Denition 1.1. ([6]) A ring R is said to be UU if U(R) = 1 +Nil(R).2010 Mathematics Subject Classication. 16D 60; 16S 34; 16U 60.Key words and phrases. Exchange rings, n-UU Rings, -UU Rings, Boolean rings,Tripotent rings.12 Peter V. DanchevDenition 1.2. A ring R is said to be exchange if, for each r 2 R, thereis an idempotent e 2 rR such that 1  e 2 (1  r)R.It was proved in [6] that a ring R is an exchange UU ring if, and only if,J(R) is nil and R=J(R) is Boolean.Before proceed by proving our chief result, we need a few more tech-nicalities, mainly developed by the current author in the papers cited inthe reference list. And so, generalizing Denition 1.1, one can state thefollowing.Denition 1.3. Let n 2 N be xed. A ring R is called n-UU if, for anyu 2 U(R), un 2 1 + Nil(R), that is, the inclusion Un(R)  1 + Nil(R)holds. If n is the minimal natural with this property, R is just said to bestrongly n-UU.Clearly, UU rings just coincide with (strongly) 1-UU rings.This can be freely expanded to the following:Denition 1.4. A ring R is called -UU if, for any u 2 U(R), there existsi 2 N depending on u such that ui 2 1 +Nil(R).The leitmotif of the present paper is to study exchange n-UU rings in thecases n = 2 and n = 3. Our results will considerably strengthen those from[1] and will also provide the interested reader with new simpler proofs. Inclosing we state a question which remains unanswered.2. Main ResultsThe next statement considerably supersedes [1, Lemma 4.4] by droppingo the unnecessary limitation on the ring to be "exchange". The usedtechnique was developed in [4] and [5].Proposition 2.1. Let R be a 2-UU ring. Then J(R) is nil.Proof. Given x 2 J(R), it follows that (1+x)2 = 1+2x+x2 2 1+Nil(R)which amounts to 2x + x2 2 Nil(R). Similarly, replacing x by  x, wederive that  2x+ x2 2 Nil(R). Since these two sums commute, it followsOn exchange -UU unital rings 3immediately that 2x2 2 Nil(R). Finally, using the above trick for x2, wededuce that 2x2 + x4 2 Nil(R). Since 2x2 2 Nil(R), we conclude thatx4 2 Nil(R), i.e., x 2 Nil(R), as required.Corollary 2.2. A ring R is 2-UU if, and only if, J(R) is nil and R=J(R)is 2-UU.Proof. According to Proposition 2.1, the argument follows in the samemanner as [6, Theorem 2.4 (2)].Lemma 2.3. Let R be a ring. Then the following two points hold:(i) If R is n-UU for some n 2 N, then eRe is also n-UU for any e 2Id(R).(ii) If R is -UU, then eRe is also -UU for any e 2 Id(R).Proof. We shall show the validity only of (ii). The proof of (i) is analogousand so it will be omitted. As in [6], letting w 2 U(eRe) with inverse v,it follows that w + 1   e 2 U(R) with inverse v + 1   e. Therefore, thereexists i 2 N such that (w + 1   e)i = wi + 1   e 2 1 + Nil(R), that is,wi   e = q 2 Nil(R). But q 2 Nil(R) \ (eRe) = Nil(eRe) which leads towi = e+ q 2 1eRe +Nil(eRe), as expected.Lemma 2.4. For any n 2 N and any non-zero ring R the full matrix ringMn(R) is not 2-UU.Proof. Since M2(R) is isomorphic to a corner ring of Mn(R) for n  2,in view of Lemma 2.3 it suces to establish the claim for n = 2. To thatgoal, as in [6], let us consider the invertible matrix0 11 1with the inverse 1 11 0. Since0 11 12=1 11 2, we infer that1 11 2 1 00 1=0 11 1which is the same invertible element with the inverse 1 11 0,and thus it is certainly not a nilpotent, as wanted.We shall now restate and reproof the main result from [1] by givinga more convenient form and more transparent proof arising from well-known recent results in [6] and [5], respectively. Actually, a new substantialachievement, including new points with more strategic estimations, arisesas follows:4 Peter V. DanchevTheorem 2.5. Suppose that R is a ring. Then the following ve items areequivalent:(a) R is exchange 2-UU.(b) J(R) is nil and R=J(R) is commutative invo-clean.(c) J(R) is nil and R=J(R) = BC, where B Q Z2 and C Q Z3for some ordinals  and .(d) J(R) is nil and R=J(R) is tripotent.(e) J(R) is nil and R=J(R)  Q Z2 Q Z3 for some ordinals  and.Proof. The equivalence (b) () (c) is exactly [5, Corollary 2.17], whereasthe equivalence (d) () (e) is obvious.We shall show that (a) () (b) is valid. To prove the left-to-rightimplication, we rst consider the semi-primitive case when J(R) = f0g.Imitating the basic idea from the proof of [6, Theorem 4.1], we arrive atthe case when eRe =M2(T ) for some idempotent e 2 R and some non-zeroring T depending on R, provided Nil(R) 6= f0g. However, with Lemma 2.3at hand we deduce that eRe is 2-UU, while with the aid of Lemma 2.4this property does not hold for M2(T ). This contradiction substantiatesthat R is reduced, i.e., Nil(R) = f0g and thus abelian. Hence R is cleanwith U2(R) = f1g which allows us to conclude with an appeal to [5] thatR is abelian invo-clean and so commutative invo-clean. Suppose now thatJ(R) 6= f0g. The fact that J(R) is nil follows directly from Proposition 2.1.Owing to [8] and Corollary 2.2, one sees that R=J(R) is exchange 2-UU,and so by what we have just already shown so far, the factor-ring R=J(R)has to be commutative invo-clean, as asserted.As for the right-to-left implication, it follows immediately by virtue of [8]that R is an exchange ring. That R is a 2-UU ring follows like this: Usingthe isomorphisms U(R)=(1+J(R)) = U(R=J(R)) = U(B)Q U(Z3), weso have U2(R=J(R)) = f1g. Furthermore, for any u 2 U(R) it must bethat u+J(R) 2 U(R=J(R)) and hence (u+J(R))2 = u2+J(R) = 1+J(R)which means that u2   1 2 J(R)  Nil(R), as required.The implication (c) ) (d) is elementary. What remains to illustrate isthe truthfulness of the implication (d) ) (a). Since tripotent rings areOn exchange -UU unital rings 5always exchange, the application of [8] shows that R is exchange. On theother side, since U(R=J(R)) = U(R)=(1 + J(R)) and U2(R=J(R)) = f1g,as shown above it follows that R is a 2-UU ring, thus completing the proofafter all.The next construction manifestly demonstrates that the theorem is nolonger true for n-UU rings when n > 2.Example 2.6. Consider the full matrix 2  2 ring R = M2(Z2). It wasproved in [2] that R is nil-clean and hence exchange. Moreover, R is a3-UU ring. However, it is easily checked that J(R) = f0g and that R iseven not tripotent (whence it is not boolean). In fact, U(R) has 6 elementssatisfying the following identities:0 11 03=0 11 0, so that0 11 0 1 00 1=1 11 1with1 11 12=0 00 0.1 00 13=1 00 1, so that1 00 1 1 00 1=0 00 0.1 10 13=1 00 1, so that1 00 1 1 00 1=0 00 0.1 01 13=1 01 1, so that1 01 1 1 00 1=0 01 0with0 01 02=0 00 0.1 11 03=1 00 1, so that1 00 1 1 00 1=0 00 0.0 11 13=1 00 1, so that1 00 1 1 00 1=0 00 0.Therefore, in all cases, U3(R)   1  Nil(R) and besides there are u 2U(R) such that u3 = 1 6= u. This concludes our claim and thus the exampleends.6 Peter V. DanchevWe nish o our work with the following question of some interest andimportance. Recall that a ring R is termed -Boolean if, for any r 2 R,there is i 2 N, which depends on r, with ri = r2i.Problem 2.7. Does it follow that R is an exchange -UU ring if, and onlyif, J(R) is nil and R=J(R) is -Boolean?If yes, this will resolve the basic problem from [3] in the armative.Acknowledgements. The author is grateful to the specialist referee andto the managing editor, Professor Takashi Koda, for their valuable serviceon the article.References[1] M. Abdolyouse and H. Chen, Rings in which elements are sums oftripotents and nilpotents, J. Algebra Appl. (3) 17 (2018), 11 pp.[2] S. Breaz, G. Calugareanu, P. Danchev and T. Micu, Nil-clean matrixrings, Linear Algebra Appl. 439 (2013), 3115{3119.[3] P.V. Danchev, A new characterization of boolean rings with identity,Irish Math. Soc. Bull. 76 (2015), 55{60.[4] P.V. Danchev, Weakly UU rings, Tsukuba J. Math. 40 (2016), 101{118.[5] P.V. Danchev, Invo-clean unital rings, Commun. Korean Math. Soc.32 (2017), 19{27.[6] P.V. Danchev and T.Y. Lam, Rings with unipotent units, Publ. Math.Debrecen 88 (2016), 449{466.[7] T.Y. Lam, A First Course in Noncommutative Rings, Second Edi-tion, Graduate Texts in Math., Vol. 131, Springer-Verlag, Berlin-Heidelberg-New York, 2001.On exchange -UU unital rings 7[8] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer.Math. Soc. 229 (1977), 269{278.Peter V. DanchevDepartment of Mathematics, University of Plovdiv, Bulgariae-mail: pvdanchev@yahoo.com(Received March 6, 2017)(Revised July 4, 2017)

On exchange π-UU unital rings

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On exchange π-UU unital rings

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