PIECEWISE SEMIPRIME RINGS AND SOME APPLICATIONS

We define piecewise semiprime (PWSP) rings R in terms of a set of triangulating idempotents in R. The class of PWSP rings properly contains both the class of semiprime rings and the class of piecewise prime rings. The PWSP property is Morita invariant and it is shared by some important ring extensions. A ring is PWSP if and only if it has a generalized upper triangular matrix representation with semiprime rings on the main diagonal. Another characterization of PWSP rings involves a generalization of the concept of m-systems and is similar to the description of a semiprime ring in terms of the prime radical. Finally we use the PWSP property to determine (right) weak quasi-Baer rings. These are rings in which the right annihilator of every nilpotent ideal is generated as a right ideal by an idempotent. Mathematics Subject Classification 2010: 16P60, 16P99, 16S50


Introduction
All rings are associative and R denotes a ring with unity 1. The word ideal without the adjective right or left means two sided ideal. A ring R is quasi-Baer (Baer) if the right annihilator of every right ideal (nonempty subset) of R is generated as a right ideal by an idempotent. We now recall a few definitions and results from [1] which motivated our study and serve as the background material for the present work. An idempotent e ∈ R is a lef t semicentral idempotent if exe = xe, for all x ∈ R. Similarly right semicentral idempotent can be defined. The set of all left (right) semicentral idempotents of R is denoted by S l (R) (S r (R)). An idempotent e ∈ R is semicentral reduced if S l (eRe) = {0, e}. If 1 is semicentral reduced, then R is called semicentral reduced. An ordered set {e 1 , ..., e n } of nonzero distinct idempotents of R is called a set of lef t triangulating idempotents of R if all the following hold.
From part (3) of the above definition, it can be seen that a set of left triangulating idempotents is a set of pairwise orthogonal idempotents. A set E = {e 1 , ..., e n } of left triangulating idempotents of R is complete, if each e i is semicentral reduced. A (complete) set of right triangulating idempotents is defined similarly. The cardinalities of complete sets of left triangulating idempotents of R are the same and is denoted by τ dim(R) [1,Theorem 2.10]. According to [1,Proposition 1.3] R has a (complete) set of left triangulating idempotents if and only if there exists an isomorphism of rings between R and an n by n upper triangular matrix ring with the (i, j)-entry R ij where each R ii is a (semicentral reduced) ring with identity, and each R ij is a left R ii right R jj bimodule for i < j. In this case, the ring R is said to have a (complete) generalized triangular matrix representation [3]. Following [1, p. 591], a ring R is called piecewise prime (abbreviated PWP) if there exists a complete set of left triangulating idempotents E = {e 1 , ..., e n } of R such that xRy = 0 implies x = 0 or y = 0 where x ∈ e i Re j , and y ∈ e j Re k for 1 ≤ i, j, k ≤ n (in this case, we say that R is PWP with respect to E). In [1,Theorem 4.11], it is shown that if R is a PWP ring then R is PWP with respect to any complete set of left triangulating idempotents of R; furthermore if τ dim(R) = n, then R is a PWP ring if and only if it is a quasi-Baer ring. The factor ring of a quasi-Baer ring by its prime radical is considered in [3]. For a comprehensive study of these concepts and results the reader is referred to [1][2][3][4][5]. It should be mentioned that in [7], modules whose endomorphism rings are of finite triangulating dimension are thoroughly investigated and the triangulating dimension is generalized from rings to modules. In this paper, we define (in Section 2 to follow) piecewise semiprime (abbreviated PWSP) rings with respect to sets of left triangulating idempotents. We show that if R is PWSP with respect to a set of left triangulating idempotents of R, then it is PWSP with respect to any complete set of left triangulating idempotents of R. We prove that R is PWSP if and only if R has a generalized triangular matrix representation with semiprime rings on the main diagonal. The PWSP property is Morita invariant and it is shared between a ring and some of the more important extensions of that ring. Another characterization of PWSP rings is given using the concept of generalized m-systems with respect to sets of triangulating idempotents. Accordingly, a ring R is PWSP with respect to a set of triangulating idempotents E if and only if for every e ∈ E, eN e = 0 where N is the prime radical of R. In the final section of this paper the concept of weak quasi Baer ring is defined and investigated in terms of PWSP property. Throughout the paper, we shall only deal with rings of finite triangulating dimension in all results.

Piecewise Semiprime Rings
In this section, we define a piecewise semiprime (ideal) ring and show that this property is shared between a ring and some of the more important extensions of that ring. Also, piecewise semiprime endomorphism rings are investigated. We begin with the following definitions.
Definition 2.1. Let I be a proper ideal of a ring R.
(1) The ideal I is called piecewise prime (PWP ideal) if there is a complete set of left triangulating idempotents E = {e 1 , · · · , e n } such that xRy ⊆ I implies x ∈ I or y ∈ I, where x ∈ e i Re j , and y ∈ e j Re k for 1 ≤ i, j, k ≤ n. (2) If E = {e 1 , · · · , e n } is a set of left triangulating idempotents then the ideal I is called a piecewise semiprime (PWSP) ideal with respect to E if for any x ∈ R and i = 1, · · · , n, Corollary 2.4. Let E = {e 1 , ..., e n } be a set of left triangulating idempotents of R. If R is PWSP with respect to E, then it is PWSP with respect to any complete set of left (right) triangulating idempotents of R.
Proof. This is evident by Theorem 2.3. Proof. If R is semiprime then each semicentral left idempotent is central. Since for each 1 ≤ i ≤ n, c i = e 1 + ... + e i ∈ S l (R) by [1, Proposition 1.6], we deduce that for Proposition 2.6. Let E = {e 1 , ..., e n } be a set of left triangulating idempotents and I be an ideal of R. Then the following are equivalent.
(1) I is a PWSP ideal with respect to E.
(2) If J is any ideal of R such that Je i J ⊆ I for some e i ∈ E, then e i Je i ⊆ I.
The proof is similar to the proof of (1) ⇔ (3).
Proposition 2.7. Let R ⊆ S be rings, R S be free with a multiplicatively closed basis set X such that rx = xr for all x ∈ X and r ∈ R. If E = {e 1 , ..., e n } is a set of left triangulating idempotents of R, then Proof. By hypothesis aRb = 0 if and only if aSb = 0 where a, b ∈ R. This proves (1). Also, if (e i se i )S(e i se i ) = 0 for some e i ∈ E, s ∈ S and s = m j=1 r j x j with x 1 , ..., x m ∈ X, then since X is a multiplicatively closed set, we can deduce that (e i r j e i )R(e i r j e i ) = 0 for each j = 1, ..., m, the proof is complete.
Suppose that R is a ring. The n by n matrix ring on R will be denoted by M n (R). Let X be a non empty set of not necessarily commuting indeterminates on R and x, y ∈ X. By R[[x]] and R < X > we mean the ring of formal power series in x and the free R-ring generated by X, respectively. Also if xy = yx and I = the ideal generated by xy − 1 in R[x, y], then the ring R[x, y]/I, as usual, we be denoted by R[x, x −1 ]; see [8, 1.2, p. 6] for an excellent reference for these rings.
., e n } be a set of left triangulating idempotents of R, x ∈ X be a non empty set of not necessarily commuting indeterminates on R, and I m be an identity matrix of size m. Then the following statements are equivalent.
Thus the equivalence (i) ⇔ (ii) is true by Proposition 2.6 and the fact that any ideal of M m (R) has the form M m (J) for some ideal J of R. The proof is now completed by Proposition 2.7. Proof. It is easy to see that for each 1 ≤ i ≤ n, e i and f i are idempotents with e i = 1 R and f i = 1 S . Thus [1, Lemma 1.2], implies that E 0 and F 0 are sets of left triangulating idempotents of R and S respectively. Proof. Let R and S be PWSP with respect to the sets of left triangulating idem- , then R, and S are PWSP with respect to E 0 and F 0 as in 2.9. To see this let Proof. For the PWSP case use induction and Proposition 2.10. The PWP case has a routine argument.
Examples and Remarks 2.12. (1) If R 1 and R 2 are PWP rings, then by Corollary 2.11, R 1 ⊕ R 2 is a PWP ring but not a prime ring.
with a nonzero bimodule A M B , then R is never a semiprime ring but it is a PWSP ring provided that A, B are so (Proposition 2.10). Also if A is a prime ring, X is a non-trivial ideal of A and R = A A/X 0 A , then R is PWSP with respect to the complete set of left triangulating idempotents This shows that R is not quasi-Baer. Hence by [1,Theorem 4.11], R is not PWP.
(3) A subring of a PWSP ring need not be PWSP. To see this let R be a PWSP ring with respect to a set of left triangulating idempotents E = {e 1 , ..., e n }. Then (4) Part (2)    Let M be a nonzero R-module.
Proposition 2.14. Let M be an R-module and S = End R (M ). Then the following statements are equivalent.
Hom R (X, M 1 ) and C = End R (X). Thus the result follows by Proposition 2.10, using an induction argument on n.
Corollary 2.15. The following statements are equivalent.
(2) There exists a set of left triangulating idempotents E = {e 1 , ..., e n } of R such that for each i, e i Re i is semiprime. (3) R has a triangular matrix ring representation with semiprime rings on the main diagonal.
Proposition 2.16. If R is a PWSP ring and e an idempotent in R, then eRe is a PWSP ring.
Proof. Let R be a ring and e an idempotent in R. Suppose that R is a PWSP ring. By Corollary 2.15, there exists a ring isomorphism φ from R to a generalized triangular matrix ring T with semiprime rings on its main diagonal. Since φ(e) is an idempotent in T then its main diagonal elements are idempotent and the restriction of φ is a ring isomorphism from eRe onto φ(e)T φ(e), where the rings on the main diagonal of φ(e)T φ(e) are also semiprime. Thus again by Corollary 2.15, eRe is PWSP.
Theorem 2.17. The property PWSP is Morita invariant.
Proof. If R is Morita equivalent to a ring S, then it is known that S is isomorphic to the ring eM n (R)e for some natural number n and suitable idempotent e in M n (R). The result is now obtained by Propositions 2.8(2) and 2.16.
Proposition 2.18. Let R be a PWSP ring with respect to a set of left triangulating idempotents E = {e 1 , ..., e n }.
sum of semiprime rings and there is a ring isomorphism such that each B i is a semiprime ring, B ij is a left B i -right B j bimodule, and k + l = n.
Proof. Let {a 1 , ..., a k } be the set of central idempotents of E. Since e = a 1 +...+a k is a central idempotent, It can be seen that {b 1 , ..., b l } = E\{a 1 , ..., a k } is a set of left triangulating idempotents for B. Thus there is an isomorphism as in ( * ) such that for each 1 ≤ j ≤ l, B j is semiprime. Also k + l = n.
Let Q be a ring. A right order in Q is any subring R such that (1) Every regular element b of R (i.e., r.ann (2) Every element of Q has the form ab −1 for some a ∈ R and some regular b ∈ R.

Piecewise m-Systems
It is common knowledge that m-systems play a useful role in the study of prime and semiprime rings. A nonempty subset S of R is called m-system if for every x and y in S there exists r ∈ R such that xry ∈ S. It is well known that for every ideal of R, the set {r ∈ R | every m-system containing r meets I} is equal to √ I := ∩P where P is the set of all prime ideals of R containing I. The purpose of this section is to develop an analogous concept for the study of PWSP rings. Definition 3.1. Let R be a ring and E = {e 1 , ..., e n } be a set of left triangulating idempotents of R. A nonempty subset S of R is a piecewise m-system with respect to E, if for each e i ae j and e j be k ∈ S, where e i , e j , e k ∈ E and a, b ∈ R, there exists r ∈ R such that e i ae j re j be k ∈ S. The intersection of all PWP ideals of R containing I will be denoted by Lemma 3.2. Let R be a ring and E be a complete set of left triangulating idempotents of R.
(1) If P is an ideal of R then R \ P is a piecewise m-system with respect to E, if and only if P is a PWP ideal. (2) Let P be an ideal of R such that e i P e i ∩ S = ∅ for some e i ∈ E and some piecewise m-system S. If P is maximal with respect to this property then P is prime. Proof.
(1) It is easy to check.
(2) Let I 1 and I 2 be ideals such that I 1 I 2 ⊆ P and a j ∈ I j \ P (j = 1, 2). Since P is maximal with respect to the mentioned property, there exist s j ∈ (e i (P + (a j ))e i ) ∩ S (j = 1, 2). Thus e i s 1 e i re i s 2 e i ∈ (e i P e i ) ∩ S which is a contradiction. Thus P is prime.
The following example shows that m-systems and piecewise m-systems are different in general. Proof. Suppose that e i re i ∈ I p . We shall show that e i re i ∈ p √ I. If there exists a PWP ideal P of R containing I such that e i re i / ∈ P , then e i re i ∈ R \ P := S and S ∩ e i Ie i = ∅. On the other hand, S is a piecewise m-system by Lemma 3.2 and by definition of I p , S meets e i Ie i , a contradiction. Thus e i re i ∈ P . This shows that Let e i re i / ∈ I p . Then there exists a piecewise m-system T such that e i re i ∈ T and T ∩ e i Ie i = ∅. Let A = {J R | I ⊆ J and e i Je i ∩ T = ∅}. By Zorn's Lemma, A has a maximal member Q. By Lemma 3.2, Q is a prime ideal of R. Thus e i re i / ∈ √ I.
Theorem 3.5. Let E = {e 1 , ..., e n } be a set of left triangulating idempotents of a ring R. Then for any ideal I in R the following are equivalent.
(1) I is a PWSP ideal with respect to E.
(2) For each e i ∈ E, e i Ie i = e i √ Ie i where √ I is the prime radical of I.  (2), Proposition 3.6. Let P be a PWP (resp. PWSP) ideal of R with respect to a complete set of left triangulating idempotents E = {e 1 , ..., e n }. Then P contains a minimal PWP (resp. PWSP) ideal with respect to E.
Proof. Let A be the set of PWP (resp. PWSP) ideals with respect to E which are contained in P . By Zorn's Lemma it can be seen that A has a minimal member which is a PWP (resp. PWSP) ideal. Thus P contains a minimal PWP (resp. PWSP) ideal.

Weak Quasi-Baer Rings
As mentioned in Section 1, for a ring R of finite triangulating dimension, it is shown in [1,Theorem 4.11] that R is PWP if and only if R is quasi-Baer. This important result naturally motivates the following question. What modification of the concept of a Baer ring is equivalent to the PWSP property? Although we do not know the precise answer to the above question, here we introduce the concept of right weak quasi Baer rings and then find a description of these rings in terms of the PWSP property. If M R is a module and X is a nonempty subset of M , then the right annihilator of X in R is denoted by r R (X). Similarly, the left weak quasi-Baer property for a ring R is defined (i.e. the left annihilator of every nilpotent ideal of R is generated as a left ideal by an idempotent). Examples 4.8 (3) shows that the weak quasi-Baer property is not left-right symmetric.
We recall from the introduction that τ dim(R) is assumed to be finite in the following results.
Proposition 4.2. If R is a right weak quasi-Baer ring, then R is PWSP.
Proof. Let E = {e 1 , ..., e n } be a complete set of left triangulating idempotents of R and let e i xe i Re i xe i = 0 for some i and element x ∈ R. Then I = Re i xe i R is an ideal of R such that I 2 = 0. Since R is right weak quasi-Baer, r R (I) = f R for some nonzero idempotent f ∈ R. Since f R is an ideal in R, zf = f zf for all z ∈ R. It follows that e i f e i is an idempotent element in e i Re i , and since e i y = ye i = y for all y ∈ e i Re i , we can deduce that e i f e i ∈ S l (e i Re i ). Therefore We are now going to find conditions under which a PWSP ring is right weak quasi-Baer. Since a PWSP ring is isomorphic to a formal triangular matrix ring  (1) T is right weak quasi-Baer.
Motivated by Proposition 4.3, we give the following definition.   (1) R is right weak quasi-Baer.
(2) R is PWSP and right e-weak quasi-Baer for each e ∈ S l (R).
(3) R is PWSP and right e-weak quasi-Baer for some reduced idempotent e in S l (R).
Proof.   (1) Any semiprime ring is clearly right weak quasi-Baer. Further examples can be readily constructed by taking S to be a semisimple ring in Corollary 4.4.
(2) Let F be a field, F n = F for n = 1, 2, ... and let F n | (a n ) ∞ n=1 is eventually constant} (3) The ring R = Z Z 2 0 Z 2 is right weak quasi-Baer, but it is not left weak quasi-Baer because the left annihilator of 0 Z 2 0 0 is not a direct summand of R R.
We end by recording a noteworthy fact: Remark 4.9. Being right weak quasi-Baer is a Morita invariant for rings. To see this, in [5, Lemma 2], let I be a nilpotent ideal.