1 Introduction

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Key words: Involution- Baer and Quasi Baer *-rings- *-Biregular *-rings- *-IFP *-rings. 2010 AMS Mathematics ... right ideals is complete (see [2], [3], [5] and [13]).
East-West J. of Mathematics: Vol. 16, No2 (2014) pp. 182-192

ON BIREGULAR, IFP AND QUASI-BAER

∗-RINGS Usama A. Aburawash and Muhammad Saad Dept. of Math. and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt e-mail: [email protected]; [email protected] Abstract The property of a quasi-Baer ∗-ring is extended to its ∗-corner. The question when a ∗-ring is quasi-Baer has particular answers. Moreover, an Abelian (∗-Abelian) Baer ∗-ring is a reduced quasi-Baer ∗-ring and vice versa. Next, ∗-biregularity is defined for rings with involution in a way consistent with the category of ∗-rings, and the relation with biregularity is given; biregularity implies ∗-biregularity while the converse is true only for proper involution. The center of such rings is shown to be ∗-biregular, too. Moreover, a necessary condition is given for a ∗-biregular ∗-ring to be quasi-Baer. Nevertheless, for a ∗-biregular ∗-ring R, some equivalences concerning the Boolean ∗-ring of central projections of R and the center of R are given. Finally, we define ∗-IFP ∗-rings and give some equivalences for this definition. The idempotents of such rings are central projections. The relation between IFP and ∗-IFP is also studied for ∗-rings (when ∗ is semiproper or the ring is Rickart). Finally, a quasi-Baer ∗-ring satisfying ∗-IFP is a reduced Baer ∗-ring.

1

Introduction

Throughout this paper all rings are associative with unity. A ∗-ring R will mean a ring with involution and a self adjoint subset S of R; that is S ∗ = S, is called a ∗-subset. A self adjoint idempotent e of R is said to be a projection; that is e2 = e = e∗ . A self adjoint element of R is called symmetric. The center and the sets of central idempotents and central projections of R will be denoted by Z(R), B(R) and B∗ (R), respectively. The right annihilator of a subset X of R is denoted by rR (X) or r(X) if there is no ambiguity. A ring (resp. ∗-ring) R Key words: Involution- Baer and Quasi Baer *-rings- *-Biregular *-rings- *-IFP *-rings. 2010 AMS Mathematics Subject Classification: Primary 16W10 Secondary 16D25, 16S15

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is said to be Abelian (resp. ∗-Abelian) if all its idempotents (resp. projections) are central. R is reduced if it has no nonzero nilpotent elements. An involution ∗ is called proper (resp. semiproper ) if aa∗ = 0 (resp. aRa∗ = 0) for every nonzero element a of R. Obviously, a proper involution is semiproper. Following [13], a ring (resp. ∗-ring) R is called a Baer ring (resp. a Baer ∗-ring), if the right annihilator of every nonempty subset of R is generated, as a right ideal, by an idempotent (resp. a projection). These definitions are leftright symmetric. However, the study of Baer rings has its roots in functional analysis; in [13] Kaplansky introduced Baer rings to abstract various properties of von Neumann algebras and complete regular ∗-rings. The class of Baer rings includes the von Neumann algebras (for example, the algebra of all bounded operators on a Hilbert space is a Baer ∗-ring under the involution of adjoint of operator), the commutative C ∗ -algebras C(T ) of continuous complex valued functions on a Stonian space T , and the regular rings whose lattice of principal right ideals is complete (see [2], [3], [5] and [13]). A generalization of Baer rings is pp-rings. A ring R is called a right (resp. left) pp-ring if the right (resp. left) annihilator of every element of R is generated, as a right (resp. left) ideal by an idempotent. R is called a pp-ring (called also a Rickart ring) if it is both a right and left pp-ring. A ∗-ring R is called a Rickart ∗-ring if the right annihilator of every element is generated, as a right ideal, by a projection. In [12], Clark defines a ring to be quasi-Baer if the right annihilator of every right ideal is generated, as a right ideal, by an idempotent (equivalently, the right annihilator of every ideal is generated, as a right ideal, by an idempotent). This principal right ideal is clearly an ideal of R; since if I is a right ideal of R and r(I) = eR with an idempotent e, then IReR ⊆ IeR = 0 implies ReR ⊆ eR. Recall from [5], an idempotent e ∈ R is left (resp. right) semicentral in R if eRe = Re (resp. eRe = eR). Equivalently, an idempotent e ∈ R is left (resp. right) semicentral in R if eR (resp. Re) is an ideal of R. Moreover, if R is semiprime then every left (resp. right) semicentral idempotent is central. The concept of quasi-Baer, however, is given to characterize when a finite-dimensional algebra with unity over an algebraically closed field is isomorphic to a twisted matrix units semigroup algebra. Moreover, it is a nontrivial generalization of the class of Baer rings. Recall from [9], a ∗-ring R is said to be a quasi-Baer ∗-ring if the right annihilator of every ideal is generated, as a right ideal, by a projection (equivalently, the right annihilator of every right ideal is generated, as a right ideal, by a projection). The class of quasi-Baer ∗-rings is, however, a nontrivial generalization of the class of Baer ∗-rings. Moreover, the property of being quasi-Baer for ∗-rings has been shown to extend to formal power series, Laurent polynomials and Laurent series over a quasi-Baer ∗-ring. Conversely, if either of these is quasi-Baer ∗-ring, then the underlying ring is also quasi-Baer ∗-ring. Finally, a ring (resp. ∗-ring) R is called regular (resp. ∗-regular ) if every principal one-sided ideal of R is generated by an idempotent (resp. a projec-

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tion). Obviously, every ∗-regular ∗-ring is regular, while the converse is true if the involution ∗ is proper. R is called strongly regular if every principal right ideal of R is generated by a central idempotent (see [15]).

2

Extending Quasi-Baer ∗-Rings

Our first result is to show that the ∗-corner of a quasi-Baer ∗-ring is also a quasiBaer ∗-ring. Remind that a quasi-Baer ∗-ring is semiprime, by Proposition 1.1 in [9]. First, we need the following lemma. Lemma 1. Let R be a semiprime ring. Then the corner eRe , for every idempotent e in R, is a semiprime ring. Proof. Let I be an ideal of the corner eRe, where e is an idempotent, then eI = Ie = eIe. If I 2 = 0, then (IR)2 = IRIR = IeReIR ⊆ I 2 R = 0 implies 0 = IR ⊇ I, from the semiprimeness of R. Thus I = 0 and eRe is semiprime. Proposition 1. Let R be a quasi-Baer ∗-ring. Then the ∗-corner eRe , for every projection e in R, is a quasi-Baer ∗-ring. Proof. Let I be an ideal of the ∗-corner T = eRe, where e is a projection. Since R is a quasi-Baer ∗-ring, then rR (IR) = fR, for some central projection f in R. Set g = ef = e2 f = efe ∈ eRe. Clearly, g is a projection in T . Since IgT = IefT ⊆ IRfR = 0, hence gT ⊆ rT (I). Conversely, let x ∈ rT (I), then Ix = 0. So (xI)2 = x(Ix)I = 0, then xI = 0 due to the semiprimeness of T , by Lemma 1. Also (IRx)2 = IR(xI)Rx = 0 implies IRx = 0. Hence x ∈ rR (IR) = fR implies x = fx = f(ex) = (ef)x = gx. So rT (I) = gT for some central projection g in T . Thus T = eRe is a quasi-Baer ∗-ring. According to Proposition 4.3 in [11], the center of a quasi-Baer ∗-ring is a Baer ∗-ring, and so it is a quasi-Baer ∗-ring. The converse of this statement is not true; we need one more condition for the converse to be true as given in the following proposition. Theorem 1. A ∗-ring R is a quasi-Baer ∗-ring if its center Z(R) is a quasiBaer ∗-ring and every nonzero ∗-ideal of R has nonzero intersection with Z(R). Proof. Put Z = Z(R). By Proposition 1.1 in [9], Z is a semiprime quasiBaer ring in which every idempotent of Z is a projection. We claim that every nonzero ideal I of R has a nonzero intersection with Z. If I ∩ I ∗ = 0, then I ∩ Z ⊇ (I ∩ I ∗ ) ∩ Z = 0, from the assumption. If I ∩ I ∗ = 0, then I has no nonzero symmetric elements. Hence, for every 0 = a ∈ I, R(a − a∗ )R is a nonzero principal ∗-ideal of R and R(a − a∗ )R ∩ Z is a nonzero ∗-ideal of Z. Therefore, there exists a nonzero central element z ∈ Z such that z = r(a − a∗ )s = ras−ra∗ s, for some r, s ∈ R. Moreover, for every x ∈ R, (ras−ra∗ s)x =

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x(ras − ra∗ s) which implies (ras)x − x(ras) = (ra∗ s)x − x(ra∗ s) ∈ I ∩ I ∗ = 0. Thus (ras)x = x(ras) and (ra∗ s)x = x(ra∗ s) which imply that ras and ra∗ s are central elements of R, which can not be zero together. Because ras ∈ I ∩ Z and ra∗ s ∈ I ∗ ∩ Z = (I ∩ Z)∗ , hence I ∩ Z = 0. Since B(Z) = B(R) and every central idempotent is semicentral, then R is a semiprime quasi-Baer ring, by ( [7], Theorem 2.2), and every central idempotent of R is a projection. Hence, R is a quasi-Baer ∗-ring, by ( [9], Proposition 1.1). In the next section, we will see that the ∗-biregular ∗-ring with center being quasi-Baer ∗-ring is a quasi-Baer ∗-ring. In the following last result of this section, we show that the Abelianity and ∗-Abelianity are equivalent for Baer ∗-rings. Moreover, this Abelian Baer ∗ring is a reduced quasi-Baer ∗-ring. By the way, it is clear from the definition that a Baer ∗-ring is a quasi-Baer ∗-ring, but the converse is not true (see [9]). Proposition 2. Let R be a ∗-ring. The following conditions are equivalent: 1. R is an Abelian Baer ∗-ring. 2. R is a ∗-Abelian Baer ∗-ring. 3. R is a reduced quasi-Baer ∗-ring. Proof. (1)⇒(2) is obvious from the definition. (2)⇒(3): R is clearly quasi-Baer ∗-ring. To show that R is reduced, let x2 = 0 for some x ∈ R, then x ∈ r(x) = eR for some central projection e in R, since R is ∗-Abelian. So that x = ex = xe = 0 and R is reduced. (3)⇒(1): R is a quasi-Baer ring and reduced, so R is an Abelian Baer ring, by ([4], Lemma 1). Moreover, the involution ∗ is proper, since aa∗ = 0 implies a∗ ∈ r(a) = eR for some central idempotent e, because R is Abelian. So that a∗ = ea∗ which gives a = ea = ae = 0. Hence R is a quasi-Baer ∗-ring, by ([11], Proposition 4.3). For commutative rings, we have the following consequence. Corollary 1. Every commutative quasi-Baer ∗-ring is a Baer ∗-ring. Proof. Let R be a commutative ∗-ring and set a2 = 0 for some a ∈ R, then aRa = 0 and hence a = 0, from the semiprimeness of R. So that R is reduced and hence a Baer ∗-ring, by Proposition 2.

3

∗-Biregular ∗-Rings

A ring R is said to be biregular if every principal ideal of R is generated by a central idempotent. Analogously, one can define a ∗-biregular ∗-ring to be such that every principal ∗-ideal of R is generated by a central projection. we note that the principal ideal a is a ∗-ideal if and only if a = a∗ . However,

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this notion of ∗-biregularity is consistent with the category of rings with involution where the objects and the morphisms should preserve the involution. Obviously, a biregular ∗-ring R is ∗-biregular, since every principal ∗-ideal I of R has the form I = eR, for some central idempotent e, whence eR = Re∗ and consequently e = ee∗ = e∗ . The following proposition answers the question when a ∗-biregular ∗-ring is biregular. Proposition 3. Let R be a ∗-biregular ∗-ring and ∗ is proper, then R is biregular. Proof. Let I be a nonzero principal ideal of R generated by a = 0. Since ∗ is proper, then Raa∗ R = 0 is a principal ∗-ideal of R. Hence, eR = Raa∗ R ⊆ I, for some central projection e of R, by the ∗-biregularity of R. Moreover, eaa∗ = aa∗ e = aa∗ implies [(1 − e)a][(1 − e)a]∗ = (1 − e)aa∗ (1 − e) = 0. Since ∗ is proper, then (1 − e)a = 0 which gives I ⊆ eR. Therefore, I = eR and R is biregular. In the following proposition, we give a necessary and sufficient condition for a ∗-ring to be ∗-regular. Proposition 4. A ∗-ring R is ∗-regular if and only if a ∈ Ra∗ a for every a ∈ R. Proof. First, let a ∈ Ra∗a for every a ∈ R. So that, there is r ∈ R such that a = ra∗ a. But (ra∗ )(ra∗ )∗ = (ra∗ a)r ∗ = ar ∗ = (ra∗ )∗ , then e = ra∗ is a projection. Hence, e = ar ∗ ∈ aR and a = ea ∈ eR imply aR = eR which means that R is ∗-regular. Conversely, let R be ∗-regular, then for every a ∈ R, aR = eR for some projection e of R. Hence a = ea and e = ar for some r ∈ R. Thus a = e∗ a = r ∗ a∗ a ∈ Ra∗ a. Next, we show that ∗-biregularity of a ∗-ring is extended to its center. Proposition 5. The center of a ∗-biregular ∗-ring is ∗-biregular. Proof. Every principal ∗-ideal of Z = Z(R) is of the form aZ = a∗ Z, and so a∗ = az, where a, z ∈ Z. Therefore, (aR)∗ = a∗ R = azR ⊆ aR. Hence aR is a principal *-ideal of R, so that there is a central projection e of R such that aR = eR. Since a = ea, we have aZ ⊆ eZ. Moreover, there is r ∈ R such that e = ar = aer. For every s ∈ R, we have (er)s = rse = rsar = arsr = esr = s(er), whence er ∈ Z and eZ ⊆ aZ. Therefore, aZ = eZ and Z is ∗-biregular. Observe that ∗-biregular ∗-rings are ”almost” quasi-Baer ∗-rings in the sense that the right annihilator of every principal ∗-ideal is generated by a projection. In general, a ∗-biregular ∗-ring is not a quasi-Baer ∗-ring as shown by the following example.

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Example 1. Let W be the first Weyl algebra over a ∗-field of characteristic zero ∞ and let R = {(xn ) ∈ n=1 W |xn is eventually constant}. Then it can be easily checked that the ∗-ring R under the componentwise involution is ∗-biregular, but not a quasi-Baer ∗-ring (see [7], example 3.1). However, if the center of a ∗-biregular ∗-ring is quasi-Baer, then the ring itself is quasi-Baer. Proposition 6. A ∗-biregular ∗-ring with center being a quasi-Baer ∗-ring, is a quasi-Baer ∗-ring. Proof. Let I be a a nonzero ∗-ideal of R. If I has a nonzero symmetric element a, then eR = RaR ⊆ I for some nonzero central projection e. Hence 0 = e ∈ I ∩ Z(R). If I has no nonzero symmetric elements, then for every 0 = a ∈ I, we have R(a − a∗ )R = fR, for some nonzero central projection f, since a = a∗ . Thus every nonzero ∗-ideal of R has a nonzero intersection with Z(R), consequently R is a quasi-Baer ∗-ring, by Theorem 1. To prove the main result for ∗-biregular ∗-rings, we need the following two lemmas. Lemma 2. Every continuous ∗-regular ∗-ring is a Baer ∗-ring. Proof. Let R be a ∗-regular ∗-ring, then R is a regular Rickart ∗-ring, by ( [16], Theorem 4.5). Since R is continuous, it is complete, by ( [15], Chapter XII, Proposition 4.10). Hence, R is a regular Baer ∗-ring, by ( [3], Corollary 1.25). Lemma 3. Let R be a quasi-Baer ∗-ring. Then the set of all principal ∗-ideals generated be central projections of R is a complete sublattice of the lattice of ideals of R. Proof. Let {eλ }λ∈Λ ⊆ B∗ (R), then there is a central projection e such that    eλ R = r(R(1 − eλ )) = r( R(1 − eλ )) = eR. λ∈Λ

λ∈Λ

λ∈Λ

This means that the set of all principal ∗-ideals generated be central projections of R is complete. Theorem 2. Let R be a ∗-biregular ring. Then the following conditions are equivalent: 1. R is a quasi-Baer ∗-ring. 2. The lattice of principal ∗-ideals of R is complete and ∗ is proper on Z(R). 3. The Boolean ∗-ring B∗ (R) of central projections of R is complete and ∗ is proper on Z(R).

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4. The Boolean ∗-ring B∗ (R) of central projections of R is self-injective and ∗ is proper on Z(R). 5. The Boolean ∗-ring B∗ (R) of central projections of R is continuous and ∗ is proper on Z(R). 6. The center Z(R) of R is a Baer ∗-ring. 7. The center Z(R) of R is continuous and ∗ is proper on Z(R). Proof. (1)⇒(2): ∗ is semiproper on R, by ( [9], Proposition 3.4), hence it is proper on Z(R), because it is commutative. Since R is ∗-biregular, then every principal ∗-ideal is generated be a central projection and the required result follows from Lemma 3, since R is quasi-Baer∗-ring. (2)⇒(3): Let {eλ }λ∈Λ ⊆ B∗ (R). Then λ∈Λ eλ R = eR for some central projection, by the completeness of the lattice of principal ∗-ideals. So that  e B (R) = eB∗ (R) and B∗ (R) is complete. λ ∗ λ∈Λ (3)⇒(4) follows from ( [15], Chapter XII, Proposition 3.3). (4)⇒(5): Clearly, B∗ (R) is strongly regular from the definition. Hence, B∗ (R) is continuous, by ( [15], Chapter XII, Proposition 4.12). (5)⇒(6): B∗ (R) is regular and hence it is complete by ( [15], Chapter XII, Proposition 4.10). Also Z = Z(R) is ∗-biregular, by Proposition 5 and hence it is biregular, by Proposition 3. The commutativity of Z makes it regular. Since ∗ is proper, then Z is a Rickart ∗-ring, by ( [16], Theorem 4.5). Let S ⊆ Z, then rZ (S) = ∩s∈S rZ (s) = ∩λ∈Λ eλ Z for some {eλ |λ ∈ Λ} ⊆ B∗ (R). Thus rZ (S) = eZ for some central projection e of R from the completeness of B∗ (R). Hence, Z is a Baer ∗-ring. (6)⇒(7): Again Z(R) is ∗-biregular, by Proposition 5 and ∗ is proper on Z(R), by ( [9], Proposition 3.1). Hence Z(R) is biregular, by Proposition 3 and so it is regular and strongly regular from the definition, because it is commutative. Hence Z(R) is a regular Baer ∗-ring, so it is complete, by ( [3], Corollary 1.25) and finally is continuous by ( [15], Chapter XII, Proposition 4.12). (7)⇒(1): As in the proof of (6)⇒(7), Z(R) is regular. Since ∗ is proper on Z(R), then Z(R) is ∗-regular. Because Z(R) is continuous ∗-regular, then it is a Baer ∗-ring, by Lemma 2 and consequently a quasi-Baer *-ring. Hence R is a quasi-Baer *-ring, by Proposition 6 .

4

∗-Rings with ∗-IFP

Recall from [1] (or [7]), a ring R is said to be semicommutative or to have IFP 1 if for every a ∈ R, r(a) is an ideal of R. Equivalently, r(X) is an ideal of R for all nonempty subsets X of R. One can define the analogous concept for a 1 IFP

is the first letters of ”insertion of factors property”

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∗-ring R; that is R has ∗-IFP if for every a ∈ R, r(a) is a ∗-ideal of R. Since l(a) = r(a∗ ) for ∗-rings having ∗-IFP, it follows that l(a) is also ∗-ideal. It is clear that every ∗-ring having ∗-IFP has also IFP while the converse is not always true as shown by the following example. Example 2. Let F be a field and consider the ∗-ring R = F ⊕ F , with the exchange involution (a, b)∗ = (b, a), for all a, b ∈ F . R has clearly IFP but it does not have ∗-IFP, since r((1, 0)) = (0, F ) is not a ∗-ideal. The following proposition gives some equivalences for the ∗-IFP property. Proposition 7. For a ∗-ring R, the following conditions are equivalent: 1. R has ∗-IFP 2. r(S) is a ∗-ideal of R for every subset S ⊆ R. 3. l(S) is a ∗-ideal of R for every subset S ⊆ R. 4. For every a, b ∈ R, ab = 0 implies aRb∗ = 0.

 Proof. (1)⇒(2): It is clear that for every S ⊆ R, r(S) = s∈S r(s) being the intersection of self-adjoint ideals is also self adjoint. (2)⇒(3): l(S) = r(S ∗ ) is a ∗-ideal of R. (3)⇒(4): ab = 0 implies b∗ a∗ = 0. Hence b∗ ∈ l(a∗ ) which is a ∗-ideal of R. Therefore bR ⊆ l(a∗ ) and aRb∗ = 0 follows. (4)⇒(1): Let x ∈ r(a), which is a right ideal of R, then ax = 0 implies aRx∗ = 0. Hence ax∗ = 0 which means x∗ ∈ r(a). Therefore r(a) is a self adjoint right ideal of R and consequently a ∗-ideal of R. Next, we see that all idempotents of a ∗-ring having ∗-IFP are central projections. Proposition 8. Every idempotent of a ∗-ring having ∗-IFP is a central projection. Proof. First, we show that R is ∗-Abelian. Let e be a projection in R, then (1 − e)e = 0 implies (1 − e)Re = 0, by Proposition 7. Hence e is left semicentral projection and consequently is central. Now, let e be an idempotent of R, then e(1 − e) = 0 implies eR(1 − e)∗ = 0 and e = ee∗ is a projection. The previous proposition declares that every ∗-ring having ∗-IFP is ∗Abelian. However, the converse is not always true as shown by example 2, where R is ∗-Abelian but has not ∗-IFP. By the way, if the involution is semiproper the properties IFP and ∗-IFP are identical as claimed by the following proposition. Proposition 9. Let R be a ∗-ring with semiproper involution ∗. Then the following conditions are equivalent:

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1. R has IFP. 2. R has ∗-IFP. 3. R is reduced. Proof. (1)⇒(2): Set ab = 0, then aRb = 0 from the IFP property of R. So that, for every r ∈ R, (arb∗ )R(arb∗)∗ = a(rb∗ R)br ∗a∗ ⊆ aRbr ∗a∗ = 0. Hence, aRb∗ = 0 from the semiproper of ∗ and consequently R has ∗-IFP, from Proposition 7. (2)⇒(3): Set a2 = 0, then aRa∗ = 0, where R has ∗-IFP. So that a = 0 from the semiproper of ∗. (3)⇒(1): Let b ∈ r(a) for some a ∈ R, then ab = 0 implies (ba)2 = baba = 0 and consequently ba = 0 from the reducedness of R. Hence for every r ∈ R, (arb)2 = arbarb = 0 implies aRb = 0. Thus r(a) is an ideal of R and therefore R has IF P . For Rickart ∗-rings, we can add more equivalences, namely Abelian and ∗-Abelian. Proposition 10. Let R be a Rickart ∗-ring, then the following conditions are equivalent: 1. R is reduced. 2. R is Abelian. 3. R is ∗-Abelian. 4. R has ∗-IFP. 5. R has IFP. Proof. (1), (4) and (5) are equivalent, by Proposition 9, since the involution of a Rickart ∗-ring is proper and hence semiproper. (1)⇒(2): is direct, since every reduced ring is Abelian by ([15], Proposition 12.3). (2)⇒(3): is clear form the definition. (3)⇒(4): Since R is a Rickart ∗-ring then for every a ∈ R, r(a) = eR = Re, where e is a central projection because R is ∗-Abelian. Thus r(a) is a ∗-ideal of R and R has IFP. The following is an involutive version of Theorem 2.4 in [7]; it gives sufficient conditions to transfer the property of quasi-Baer from a ∗-subring to the whole ∗-ring with more additional properties. Theorem 3. Let S be a quasi-Baer ∗-subring of a ∗-ring R having ∗-IFP such that every nonzero ∗-ideal of R has nonzero intersection with S. Then R is a right nonsingular quasi-Baer ∗-ring and S is a reduced Baer ∗-ring.

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Proof. Let I be a nonzero ideal of R, then II ∗ = 0, by ( [9], Theorem 1.8) and consequently 0 = II ∗ ∩ S ⊆ I ∩ S. Hence, every nonzero ideal of R has nonzero intersection with S. Moreover, R has ∗-IFP means that it has also IFP. According to ( [7], Theorem 2.4), R is a semiprime right nonsingular quasiBaer ring and S is a reduced Baer ring. Now, S is a reduced quasi-Baer ∗-ring implies that S is a Baer ∗-ring, by Proposition 2. Finally, by Proposition 8, every idempotent of R is a central projection. Hence R is a quasi-Baer ∗-ring, by ( [9], Proposition 1.1). Finally, a sufficient condition is given for a quasi-Baer ∗-ring to be reduced Baer. Proposition 11. Let R be a quasi-Baer ∗-ring satisfying ∗-IFP, then R is a reduced Baer ∗-ring. Proof. Let a2 = 0 for some a ∈ R, then aRa∗ = 0, by Proposition 7. But ∗ is semiproper, by ( [11], Proposition 4.3), then a = 0 and R is reduced. From Proposition 2, it follows that R is a reduced Baer ∗-ring.

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