STRONGLY CLEAN ELEMENTS IN A CERTAIN BLOCK MATRIX RING

Transcription

1 J. Indones. Math. Soc. Vol., No. 06, pp STONGLY CLEAN ELEMENTS IN A CETAIN BLOCK MATIX ING Abolfazl Nikkhah and Bijan Davvaz Department of Mathematics, Yazd University, Yazd, Iran nikkhah@stu.yazd.ac.ir, davvaz@yazd.ac.ir Abstract. In this paper, we investigate the concepts of strongly clean, strongly π-regular and strongly J n-clean related to the ring A M S. Moreover, we give several equivalent conditions such that an element α EndA is strongly clean. Key words and Phrases: Strongly clean, strongly π-regular, strongly J n-clean, endomorphism. Abstrak. Dalam makalah ini, konsep dari strongly clean, strongly π-regular dan strongly J n-clean yang berhubungan dengan ring A M S diselidiki. Selanjutnya, beberapa kondisi ekivalen diberikan sedemikian sehingga sebuat element α EndA adalah strongly clean Kata kunci:strongly clean, strongly π-regular, strongly J n-clean, endomorphism.. Introduction An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commute with each other. This notion was firstly introduced by Nicholson in [?]. Let, S be two rings, and let M be an S- bimodule. This means that M is a left -module and a right S-module such that rms rms for all r, m M, and s S. Given such a bimodule M, we can form { } A M S r m : r, m M, s S, 000 Mathematics Subject Classification: 5A3, 6L99. eceived: , revised: , accepted:

2 5 A. Nikkhah and B. Davvaz and define a multiplication on A by using formal matrix multiplication: r m r m rr rm + ms s. In [?], a characterization of strongly J n -clean rings by virtue of strongly π- regularity is given. The main purpose of this note is to study about strongly clean, strongly π-regular and strongly J n -clean of the ring A. We give several equivalent conditions under which such element α EndA is strongly clean.. Strongly clean, strongly π-regular and strongly J n -clean Let us define the following subsets of a ring. U {r r is a unit of }, ID {r r is an idempotent of }. We begin with M Proposition.. Let A, where and S are commutative rings and M S is a bimodule. Then, A is a strongly clean ring if and only if and S are strongly clean rings. r 0 r 0 r m Proof. Suppose that r, s S and A. Then, + r m such that r U, s US, r ID, s IDS, m, m M r 0 r + r and r m + m s m. So, m + m. Hence, r r + s + r and s s + s. r m For the converse, suppose that and S are strongly clean rings and r m r + r A. Then, m such that r + s U, s US, r m r m r ID and s IDS. So, r m +, where 0 s 0 s r m m +m m and r m +m s m. Hence, r m UA and IDA. Definition.. An element in a ring is called strongly π-regular if for every a, the chain a a... terminates or equivalently, the chain a a... terminates, see [?,?].

3 Strongly Clean Elements 53 M As defined by artinian ring, if A is artinian, then the chain r m r m A A... terminates. So, A is strongly π-regular. M emark. [?] Let A, where and S are rings and M S is a bimodule. Then, A is artinian if and only if and S are artinian. M Proposition.3. Let A, where and S are artinian rings and M as a S-module is artinian. Then, A is strongly π-regular. Proof. The verification is straightforward. Definition.4. We say that x is strongly J n -clean provided that there exists an idempotent e such that x e U, ex xe and ex n J, where J is the Jacobson radical of. M Proposition.5. Let A, where and S are commutative local rings r m and M S is a bimodule. Then, A is strongly J n -clean if there exists r m an idempotent A such that r m + m s rm + ms and r r n s s n 0. Proof. By 0 M 0 0 r m r m r m r m J M and JA 0 JS, the proof is clear. 3. Endomorphisms and strongly clean In this section, we study the endomorphisms of the ring A and also necessary and sufficient conditions under which an endomorphism of the ring A is strongly clean. Lemma 3.. Let A M S, α EndA and let π π EndA. Then, πa is α-invariant if and only if πα παπ. Both πa and πa are α-invariant if and only if πα απ. Proof. The proof is a routine verification.

4 54 A. Nikkhah and B. Davvaz Proposition 3.. Let A M S, α EndA and let A P S Q S, }{{}}{{} A A where A and A are both α-invariant. The following are equivalent. α A is an isomorphism. kerα A and A αa. Proof.. If F kerα write F G + H. Then, 0 αf αg + H αg + αh A A, so 0 αg α A G. Thus, we have G 0 by, so F H A. Hence, kerα A. Finally, A α A A αa αa.. α A is one-to-one because kerα A kerα A A A 0. Let π π EndA satisfy kerπ A and πa A. Then, απ πα by??, so A πa παa απa αa using. Hence, α A is onto. emark. In Proposition??, it is important the existence of a ring with the property. Here we mention a well known fact. If V is an infinite dimensional vector space over a field F, then EndV F is a ring with the property that. Theorem 3.3. Let A M S and E EndA. The following are equivalent for α E. α is strongly clean in E. There exists π π E such that απ πα, απ is a unit in πeπ and α π is a unit in πe π. 3 A P S Q S, where A and A are α-invariant, and 0 } S {{ } A 0 } S {{ } A α A and α A are isomorphisms. 4 A A A, where A and A are α-invariant, kerα A αa and ker α A αa. 5 A A A... A n for some n where A i is α-invariant and α Ai is strongly clean in EndA i for each i. Proof.. Let α π + σ where πσ σπ, π π, and σ UE. Note that and α, π, and σ all commute. Now α σ π so απ σπ. Since σ α πeπ this gives απσ π σπσ π π. Similarly, σ παπ π so απ is a unit in πeπ. Finally, observe that α π + σ is strongly clean too, so an analog of the above argument shows that α π is a unit in πe π. 3. Given π as in let A πa and A πa. Then, A A A and αa απa παa πa A, so A is α-invariant. Similarly, αa. If απ πγπ in πeπ, let γ 0 πγπ A. Then, γ 0

5 Strongly Clean Elements 55 EndA so, if F A we have γ 0 α A F γ 0 απf γ 0 απf γ 0 παf πf F. Thus γ 0 α A A. Similarly α A γ 0 A, because α A γ 0 F α A πγπf απγπf πf F. Thus α A is a unit in EndA. A similar argument shows that α A is a unit in EndA This follows from Proposition?? Suppose that A A A as in 4. Then, α A is strongly clean in EndA because it is a unit by Proposition??. Similarly A and A are α-invariant so α A is a unit in EndA, again by Proposition??. This gives 5 with n, A A and A A. 5. Given the situation in 5, extend maps λ i EndA i to λ i EndA by defining λ i F + F F n λ i F i. Then, λ i λj 0 if i j while λ i µ i λ i µ i for all µ i EndA i. Now let α Ai π i + σ i EndA i where πi π i, σ i UEndA i and π i σ i σ i π i. If π Σ i π i and σ Σ i σ i, then π Σ i π i π, πσ Σ i π i σ i Σ i σ i π i σπ, and σ UEndA because σ Σ σ i i. Since α Σ i α Ai Σ iπ + σi π + σ, the proof of is complete. A ring is called uniquely clean [?] if every element is uniquely the sum of an idempotent and a unit. Definition 3.4. An element a of a ring is called uniquely strongly clean or USC for short if a has a unique strongly clean expression in as stated above. The ring is called uniquely strongly clean or USC for short if every element of is uniquely strongly clean. Proposition 3.5. Let A M S and α EndA. The following are equivalent: α is USC in EndA. There exists a unique decomposition A A A where A and A are α-invariant, and α A and α A are isomorphisms. Conclusion. A ring is said to be clean if every element of A can be written as a sum of an idempotent and a unit. Till now, many authors considered clean rings and obtained many results in this respect. In this work, we investigated the ring A M S. In particular, we studied the endomorphisms of the ring A and gave necessary and sufficient conditions under which an endomorphism of A is strongly clean. eferences [] Azumaya, G., Strongly π-regular rings, J. Fac. Sci. Hokkiado U., 3 954,

6 56 A. Nikkhah and B. Davvaz [] Chen, J., Wang, Z. and Zhou, Y., ings in which elements are uniquely the sum of an idempotent and a unit that commute, J. Pure Appl. Algebra, 3 009, 5-3. [3] Chen, H., Some classes of strongly clean rings, Bull. Iran. Math. Soc., 30:6 03, [4] Nicholson, W.K., Strongly clean rings and Fitting s lemma, Comm. Algebra, 7:8 999, [5] Lam, T.Y., A First Course in Noncommutative ings, Springer-Verlag, 99. [6] ajeswari, K.N. and Aziz,., A note on clean matrices in M Z, Int. J. Algebra, 3: , 4-48.