Vol. 20 (2019), No. 2, pp. 635–650 DOI: 10.18514/MMN.2019.2676
REVERSIBLE AND REFLEXIVE PROPERTIES FOR RINGS WITH INVOLUTION
USAMA A. ABURAWASH AND MUHAMMAD SAAD Received 22 September, 2018
Abstract. In this note, we give a generalization for the class of *-IFP rings. Moreover, we in- troduce *-reversible and *-reflexive *-rings, which represent the involutive versions of reversible and reflexive rings and expose their properties. Nevertheless, the relation between these rings and those without involution are indicated. Moreover, a nontrivial generalization for *-reflexive
*-rings is given. Finally, in *-reversible *-rings it is shown that each nilpotent element is *- nilpotent and K¨othe’s conjecture has a strong affirmative solution.
2010Mathematics Subject Classification: 16W10; 16N60; 16D25
Keywords: involution, quasi-*-IFP, *-reduced, *-nilpotent, *-reversible, *-reflexive and projec- tion *-reflexive
1. INTRODUCTION
All rings considered are associative with unity. A *-ring R will denote a ring with involution and a self-adjoint ideal I of R; that is IDI, is called *-ideal.
A projection e of R is an idempotent satisfies e2De De. Recall from [7], an idempotente2Ris left (resp. right) semicentral inRifeReDRe(resp.eReDeR).
Equivalently, an idempotente2R is left (resp. right) semicentral inR ifeR(resp.
Re) is an ideal of R. Moreover, if R is semiprime then every left (resp. right) semicentral idempotent is central. A semicentral projection is clearly central. A ring (resp. *-ring)R 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 for every nonzero element aof R, aaD0 (resp. aRaD0) implies aD0. Obviously, a proper involution is semiproper.
From [5],Ris semicommutativeor hasIFP if the right annihilatorr.a/D fx2 AjaxD0gof every elementa2Ris a two-sided ideal. In [1], the involutive version of IFP, that is*-IFP, is given as the ring in which the right annihilator of each element ofRis *-ideal. Clearly, each *-ring having-IFP has also IFP.
Cohn [9] called a ring R reversible (or completely reflexive) ifab D0 implies baD0for everya; b2R. Clearly, the class or reversible rings contains the reduced
c 2019 Miskolc University Press
rings. Moreover, each reversible ring has IFP. Moreover, in [9, Theorem 2.2], Cohn proved that for reversible rings, K¨othe’s conjecture has an affirmative solution. Here, we give a strong affirmative solution for K¨othe’s conjecture for *-reversible *-rings and show that each nilpotent element is *-nilpotent.
In [13], Mason introduced a generalization of reversible rings; namely reflexive rings. A right idealI of a ringRis said to bereflexiveifaRbI impliesbRaI, for everya; b2R. A ringRis calledreflexiveif0is a reflexive ideal. In [10], Kim and Baik defined anidempotent reflexiveideal as a right idealI satisfyingaReI if and only if eRaI for e2De; a 2R. R is an idempotent reflexive ring if 0 is an idempotent reflexive ideal. Obviously, the class of idempotent reflexive rings contains reflexive rings and Abelian rings.
domain +3reduced +3reversible +3 "*
reflexive +3idempotent reflexive
IFP +3Abelian
KS
A subringB of a *-ringRis said to be a*-biideal,or self adjoint biideal, ofRif BRBBandBDB.
Recall from [2], a nonzero elementaof a *-ringRis a *-zero divisorifabD0 andabD0for some nonzero elementb2R. Obviously, a *-zero divisor element is a zero divisor, but the converse is not true (example 3 in [2]). A *-ring without *-zero divisors is said to be a*-domain.
Recall from [3], an elementaof a *-ringRis said to be *-nilpotent if there exist two positive integersmandnsuch thatamD0 and.aa/nD0. Ris a *-reduced
*-ring if it has no nonzero *-nilpotent elements; equivalentlya2DaaD0implies aD0 for every a2R. A reduced (or *-domain) *-ring with proper involution is
*-reduced. Moreover, every *-reduced *-ring is semiprime.
From [4], the*-right annihilatorof a nonempty subsetSof a *-ringRis the self adjoint biidealr.S/D fx2AjSxD0DSxg. Finally, Mn.R/will denote the full matrix ring of allnnmatrices overR.
2. *-RINGS WITH QUASI-*-IFP
In this section, we introduce the property of having quasi--IFP which generalizes that of having *-IFP introduced in [1].
Definition 1. A *-ringRis said to havequasi-*-IFPif for everya2R, the *-right annihilatorr.a/is a *-ideal ofR.
In view ofl.a/Dr.a/, we see that the *-left annihilator is also *-ideal. Thus the definition of quasi-*-IFP *-ring is left-right symmetric.
Clearly, every *-ringR having *-IFP has also quasi-*-IFP, since r.a/ is *-ideal impliesr.a/Dr.a/for alla2A. However, the converse is not true as shown by the following example.
Example1. Consider the-ringRD
F F 0 F
, whereF is a field and the ad- joint of matrices is the involution. Sincer
1 0 0 0
D
0 0 0 F
is not an ideal ofR, thenRdoes not haveIFP and consequently does not have *-IFP. Moreover,R has quasi-*-IFP since the *-right annihilator of every nonzero noninvertible element ofRtakes the form
0 F 0 0
which is a *-ideal ofR.
The following are some equivalents for a *-ring to have quasi-*-IFP.
Proposition 1. For a *-ringR, the following conditions are equivalent:
(1) Rhas quasi-*-IFP.
(2) r.S/is a *-ideal ofRfor every subsetSofR.
(3) l.S/is a *-ideal ofRfor every subsetSofR.
(4) For everya; b2R,abDabD0impliesaRbD0(consequentlyaRbD0) Proof. (1))(2): For everySR, r.S/DT
s2Sr.s/ being the intersection of
*-ideals is also a *-ideal.
(2))(3): From (2),l.S/Dr.S/is a *-ideal ofR.
(3))(4):abDabD0impliesbaDbaD0and consequentlyb; b2l.a/ which is a *-ideal ofR. HencebR; bRl.a/from whichbRaDbRaD0 and thereforeaRbDaRbD0.
(4))(1): Letx2r.a/, which is a self-adjoint biideal ofR, thenaxDaxD0 impliesaRxDaRxD0, form the assumption. HenceRxr.a/which means that r.a/is a left ideal ofR. Thereforer.a/is a *-ideal due to its self-adjointness.
The following results show that quasi-*-IFP implies *-Abelian while the converse is not true.
Proposition 2. Every *-ring with quasi-*-IFP is *-Abelian.
Proof. Let e be a projection in R, then .1 e/eD.1 e/eD0 implies .1 e/Re D0, from Proposition 1. Hence e is a left semicentral projection and con-
sequently is central.
Moreover, The next example shows that the converse of Proposition2is not true.
Example 2. Let F be a field of characteristic 2 and consider the *-ring R D 8
ˆˆ
<
ˆˆ :
0 B B
@
a a12 a13 a14
0 a a23 a24 0 0 a a34
0 0 0 a
1 C C
Aja; aij 2F 9
>>
=
>>
;
, with involution defined as
0 B B
@
a a12 a13 a14
0 a a23 a24
0 0 a a34
0 0 0 a
1 C C A
D 0 B B
@
a a34 a24 a14
0 a a23 a13
0 0 a a12
0 0 0 a
1 C C A .
Since for the matrices x D 0 B B
@
0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 C C A
and y D 0 B B
@
0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
1 C C A , we havexyD0Dxy, while
x´yD 0 B B
@
0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
1 C C A
0 B B
@
a a12 a13 a14
0 a a23 a24
0 0 a a34
0 0 0 a
1 C C A
0 B B
@
0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
1 C C AD 0
B B
@
0 0 0 a23 0 0 0 0 0 0 0 0 0 0 0 0
1 C C
A¤0, for every ´2Rwith a23¤0, it follows thatR does not have quasi--IFP, by Proposition1. MoreoverRis-Abelian since for any projec- tioneD
0 B B
@
a a12 a13 a14
0 a a23 a24
0 0 a a34
0 0 0 a
1 C C A
,e2DeDe impliesa11Da12Da13Da21D a22Da33D0anda2Da, so thatRhas no nontrivial projections.
Next, we answer the question of when a *-ring with quasi-*-IFP is *-reduced.
Proposition 3. Let R be a semiprime *-ring having quasi-*-IFP, then R is *- reduced.
Proof. Let R be a semiprime *-ring having quasi-*-IFP. Set a2 DaaD0 for somea2R, thenaRaDaRaD0, from Proposition1. SinceRis semiprime, then
aD0andRis *-reduced.
Finally, one can easily show that the class of *-rings having quasi-*-IFP is closed under direct sums (with changeless involution) and under taking *-subrings.
Proposition 4. The class of *-rings having quasi-*-IFP is closed under direct sums and under taking *-subrings.
3. *-REVERSIBLE*-RINGS
Definition 2. An idealI of a *-ringRis called*-reversibleifab; ab2I implies ba2I, for everya; b2R.
It is obvious that ifI is *-reversible thenab; ab2I implies alsoba2I, for everya; b2R.
We note the following:
A one-sided *-reversible ideal must be two-sided ideal.
The *-reversible ideal may not be self adjoint according to the following example.
Example 3. LetR be the *-ring in Example1. The ideal I D
F F 0 0
is *- reversible but not self-adjoint
Definition 3. A *-ringRis said to be*-reversibleif0is a *-reversible ideal ofR;
that isabDabD0impliesbaD0(consequentlybaD0), for everya; b2R.
Example4. Every *-domain is a *-reversible *-ring.
It is clear that every reversible ring with involution is *-reversible. But the converse is not always true as shown by the next example.
Example5. LetRbe the *-ring in Example1.Ris not reversible since the matrices
˛D
0 1 0 0
andˇD
1 1 0 0
satisfy˛ˇD0whileˇ˛¤0. Moreover, it easy to check thatRis *-reversible.
The following are some equivalents for a *-ring to be *-reversible.
Proposition 5. For a *-ringR, the following statements are equivalent.
(i) Ris *-reversible.
(ii) r.S/Dl.S/for every subsetSofR.
(iii) r.a/Dl.a/for every elementa2R.
(iv) For any two nonempty subsetsAandBofR,ABDABD0impliesBAD 0(consequentlyBAD0) .
Proof. .i /).i i /: Letx2r.S/, thensxDsxD0for everys2S. SinceRis
*-reversible, we havexsDxsD0for everys2S. Hence,xSDxS impliesx2 l.S/and we getr.S/l.S/. Similarly,l.S/r.S/andr.S/Dl.S/follows.
.i i /).i i i /is direct by consideringS as the singleton setfag.
.i i i /).iv/: Set ABDABD0 for some nonempty subsets A andB ofR.
ThenabDabD0for everya2Aandb2B, and henceb2r.a/Dl.a/from the condition. ThereforebaDbaD0D0for everya2Aandb2Bwhich implies BADBAD0.
.iv/).i /is direct by consideringAandBas the singleton sets containingaand
b, respectively.
The question when does a *-reversible *-ring become reversible has been answered in the following proposition.
Proposition 6. LetRbe a *-reversible *-ring and either (1) Rhas *-IFP, or
(2) * is proper.
Then,Ris reversible.
Proof. (1) LetRhave *-IFP andabD0for somea; b2R. Then, by [1, Pro- position 7],aRbD0and henceabD0. The *-reversibility ofRimplies baD0andRis reversible.
(2) Let the involution * be proper andabD0for somea; b2R. Thena.bb/D a.bb/ D0 and hence bba D0 from the *-reversibility of R. Now .ab/.ab/DabbaD0impliesabDbaD0, since * is proper. Fi- nally, by the *-reversibility of R, baa D 0 implies aab D 0 and .ba/.ba/DbaabD0impliesbaD0. HenceRis reversible.
Now, we see that each *-reversible *-ring has quasi-*-IFP.
Proposition 7. Every *-reversible *-ring has quasi-*-IFP.
Proof. LetabDabD0for some elementsa; bof a *-reversible *-ringR. Using the *-reversibility of R, we have baDbaD0 which implies bar Dbar D0.
Again, by the *-reversibility of R, arb Darb D0 for every r 2R. Therefore aRbDaRbD0which means thatRhas quasi-*-IFP, by Proposition1.
From Propositions 7and 2, we get the following.
Corollary 1. Every *-reversible *-ring is *-Abelian.
However, the next example shows that the converse of the previous proposition and its corollary is not always true.
Example6. LetDbe a commutative domain. Then the ring RD
8
<
: 0
@
a b d 0 a c 0 0 a
1
Aja; b; c; d 2D 9
=
;
has IFP, by [11, Proposition 1.2]. Define an involution * on R as 0
@
a b d 0 a c 0 0 a
1 A
D 0
@
a c d 0 a b 0 0 a
1
A. One can easily check thatRhas quasi-*-IFP
and hence is *-Abelian. ButRis not *-reversible since the elements˛D 0
@
0 0 0 0 0 1 0 0 0
1 A
andˇD 0
@
0 1 0 0 0 0 0 0 0
1
AofRsatisfy˛ˇD˛ˇD0butˇ˛D 0
@
0 0 1 0 0 0 0 0 0
1 A¤0
Moreover, if the involution * is proper then the properties IFP, *-IFP, quasi-*-IFP,
*-reversibility and reducedness are identical as shown in the following result.
Proposition 8. LetRbe a *-ring and the involution * is proper. Then the following conditions are equivalent:
(1) Ris *-reversible (2) Rhas quasi-*-IFP.
(3) Rhas IFP.
(4) Rhas *-IFP.
(5) Ris reduced.
Proof. (3),(4) and (5) are equivalent from [1, Proposition 9].
(1))(2) is direct from Proposition7.
(2))(3): Let ab D0for some a; b2R. Then a.bb/Da.bb/D0 implies aRbbD0from the quasi-*-IFP ofR. Now.arb/.arb/DarbbraD0implies arbD0fore everyr2Rsince * is proper. ThereforeaRbD0and soRhas IFP.
(5))(1): Let ab DabD0 for some a; b2 R, then .ba/2 DbabaD0 and .ba/2DbabaD0. Hence,baDbaD0from the reducedness ofRand soR
is *-reversible.
Next, we discuss the converse of Example4; that is when a *-reversible *-ring is
*-domain.
Proposition 9. A *-ring is a *-domain if and only ifRis *-prime and *-reversible.
Proof. First, Suppose that R is a *-domain, hence R is obviously *-reversible.
Let IJ D0 for some *-ideals I and J ofR, then abDabD0for every a2I andb2J. Hence, eitheraD0orbD0which implies I D0orJ D0 and soR is *-prime. Conversely, let R be both *-prime and *-reversible andabDab D0 for some 0¤a; b2R. We have rbaDrbaD0 for every r 2R and so arbDarbD0 for every r 2R from the *-reversibility of R, which gives bRaDbRaD0. SinceRis *-prime anda¤0, we getbD0, by [ [6], Proposition 5.4], and soRhas no *-zero divisors; that is a *-domain.
As a consequence, we get Proposition 4 in [3] as a corollary.
Corollary 2 ( [3], Proposition 4). If R is a reduced *-prime *-ring, then R is
*-domain.
For a *-ring R, the trivial extension of R, denoted by T .R; R/, is the ring a b
0 a
ja; b2R
. One can define the componentwise involution
a b 0 a
D a b
0 a
to makeT .R; R/a *-ring.
Proposition 10. LetRbe a *-reduced *-ring. IfRis *-reversible, thenT .R; R/
is a *-reversible *-ring.
Proof. Let
a b 0 a
˛ ˇ 0 ˛
D
a b 0 a
˛ ˇ 0 ˛
D
0 0 0 0
. Then a˛Da˛D0andaˇCb˛DaˇCb˛D0. SinceRis *-reversible then ˛aD
˛aD0. By the *-reversibility ofR, it is easy to see that aR˛D0 . Now0D aˇCb˛D˛.aˇCb˛/D˛b˛and0DaˇCb˛Daˇ˛Cb˛˛Db˛˛. Hence .b˛/2Db˛b˛D0 and.b˛/.b˛/Db˛˛bD0. Then b˛D0becauseR is *- reduced and thereforeaˇD0. Similarly, one can show thatb˛D0andaˇD0.
Using the *-reversibility ofRagain we get˛bD˛bDˇaDˇaD0which implies ˛ ˇ
0 ˛
a b 0 a
D
˛ ˇ 0 ˛
a b 0 a
D
0 0 0 0
. ThusT .R; R/ is a
*-reversible *-ring.
Furthermore, one can easily show that the class of *-reversible *-rings is closed under direct sums (using changeless involution) and taking *-subrings.
Proposition 11. The class of *-reversible *-rings is closed under direct sums and under taking *-subrings.
4. *-REFLEXIVE*-RINGS
In this section, we introduce the involute version of reflexive ideals and rings defined by Mason [13] and study the relation between these rings and the *-reversible rings introduced in the previous section.
Definition 4. A idealI of a *-ringR is called*-reflexive if for everya; b2R, aRb; aRbI impliesbRaI (consequentlybRaI). A *-ringR is said to be*-reflexiveif0is a *-reflexive ideal ofR.
By the way, the ideal in the previous definition can not be one sided since for every a2I satisfyingaRI impliesRaI by takingbD1. Also, this ideal need not be self-adjoint by Example3.
Example7. Every *-reduced *-ring is *-reflexive.
It is evident that every reflexive *-ring is *-reflexive. However, the next example shows that the converse is not true.
Example8. LetDbe a commutative domain andRD f 0
@
˛ ˇ ı 0 ˛ 0 0 ˛
1
Aj˛; ˇ; ; ı2 Dg. R is not reflexive according to [12, Example 2.3]. Define the involution W 0
@
˛ ˇ ı 0 ˛ 0 0 ˛
1 A!
0
@
˛ ı 0 ˛ ˇ 0 0 ˛
1
A. It is easy to check thatR is-reversible and in particular is-reflexive.
Lemma 1. LetRbe a ring with semiproper involution *. ThenaRbD0implies aRbDbRaDbRaD0.
Proof.
.arb/R.arb/DarbRbraaRbraD0;
for everyr2RimpliesaRbD0;
.bra/R.bra/DbraRarbbraRbD0; for everyr2RimpliesbRaD0 and
.bra/R.bra/DbraRarbbraRbD0;for everyr2RimpliesbRaD0:
Corollary 3. Every *-ring with semiproper involution is reflexive (and hence *- reflexive).
The converse of the previous corollary is not necessary true as shown in the next example.
Example9. IfF is a field, then the ringRDF˚Fop, with the exchange involu- tion * defined by.a; b/D.b; a/for alla; b2R, is obviously a reflexive and hence
*-reflexive but * is not semiproper. Indeed, the element 0¤˛ D.0; a/ for some nonzero elementaofF satisfies˛R˛D0.
In the following proposition, we state some equivalent definitions for a *-ring to be *-reflexive .
Proposition 12. For a *-ringR, the following statements are equivalent : (i) Ris *-reflexive.
(ii) r.aR/Dl.Ra/for everya2R.
(iii) For any two nonempty subsetsAandB ofR ,ARB DARBD0 implies BRADBRAD0.
Proof. .i /).i i /: Letx2r.aR/, thenaRxDaRxD0. HencexRaDxRaD 0, by the *-reflexivity ofR, impliesx2l.Ra/and so r.aR/l.Ra/. Similarly, l.aR/r.Ra/and we getr.aR/Dl.Ra/.
.i i /).i i i /: Set ARB DARBD0 for some subsets Aand B of R. Then aRbDaRbD0for everya2Aandb2B, and henceb2r.aR/l.Ra/from the condition. ThereforebRaDbRaD0for everya2Aandb2Bwhich implies BRADbRAD0.
.i i i /).i /is direct by considering AandB as the singleton sets containing a
andb, respectively..
The following proposition and example show that the class of *-reflexive *-rings generalizes strictly that of *-reversible *-rings.
Proposition 13. Every *-reversible *-ring is *-reflexive
Proof. LetaRbDaRbD0, thenab DabD0implies rabDrabD0, for everyr2R. So thatbraDbrafor everyr2R, from the *-reversibility ofR. Thus
bRaDbRaD0and henceRis *-reflexive.
Example10. Letn > 2be an integer andpnbe a prime number. The *-ring RDMn.Zp/, where * is the transpose involution, is prime and hence reflexive (in particular *-reflexive). Moreover, R is not *-reversible. Indeed , the nonzero ele- ments
˛De12Ce13C Ce1n;
ˇDe11Ce12C Ce1.n 1/C2e1n
ofR, whereeij is the square matrix of order n with1 in the .i; j /-position and 0 elsewhere, satisfy˛ˇD˛ˇD0, whileˇ˛¤0andˇ˛¤0.
The question when a *-reflexive *-ring is *-reversible is answered in the following proposition.
Proposition 14. A *-ringR is *-reversible if and only ifRhas quasi-*-IFP and
*-reflexive.
Proof. The necessity is obvious. For sufficiency, letabDabD0for somea; b2 R. SinceRhas quasi-*-IFP, thenaRbDaRbD0. The *-reflexivity ofRimplies bRaDbRaD0. HencebaDbaD0andRis *-reversible.
In the next result we discuss when a principal right ideal generated by a projection in a *-reflexive *-ring is *-reflexive.
Proposition 15. Lete be a projection of a *-reflexive *-ringR. Theneis central if and only ifeRis a *-reflexive *-ideal.
Proof. Let e be central and aRb; aRbeR, then arb Dearb and arbD earb for everyr 2R. Hence .1 e/aRb D.1 e/aRbD0 and consequently .1 e/bRa D.1 e/bRaD0, since R is *-reflexive and e is central. Hence bRa; bRa eR andeR is *-reflexive ideal. The converse implication is clear
sinceeRis a *-ideal and soeis central.
Now, we show that *-reflexive property is extended to the *-corner.
Proposition 16. Let R be a *-reflexive *-ring, then the *-corner eRe for every projectioneofRis also *-reflexive.
Proof. LetR be *-reflexive and aDexe; bDeye2eRe such that a.eRe/bD a.eRe/bD0. ThenexeReyeDexeReyeD0implieseyeRexeDeyeRexeD 0, since R is *-reflexive. Therefore b.eRe/aDb.eRe/aD0 and so eRe is *-
reflexive.
Next, we illustrate by example that *-reflexivity is not closed under taking *- subrings.
Example11. The ringRDM2.Z2/is prime and hence reflexive. The upper trian- gular matrix ringSD
Z2 Z2
0 Z2
overZ2is a *-subring ofRunder the involution
* defined as
a b d c
D
c b d a
.Ris clearly *-reflexive butSis not, since the elements˛D
0 0 0 1
andˇD
0 1 0 0
ofRsatisfy˛RˇD˛RˇD0but ˇR˛DˇR˛D
0 Z2
0 0
¤0
We end this section by showing that the *-reflexivity is restricted from the full matrix ring to its underlying ring.
Proposition 17. If Mn.R/ is a *-reflexive *-ring for some n1 and with the transpose involution *, thenRis also a *-reflexive *-ring .
Proof. letMn.R/be a *-reflexive *-ring for somen1. SinceRŠe11Mn.R/e11,
as *-rings, thenRis *-reflexive, by Proposition16.
5. PROJECTION*-REFLEXIVE RINGS
In this last section, we give another generalization for the class of *-reflexive rings;
that is projection *-reflexive *-rings.
In [10], Kim defines an idempotent reflexive ringRas the ring satisfyingaReD0 if and only ifeRaD0for every idempotente; a2R.
Definition 5. An idealI of a *-ringRsatisfiesaReI if and only ifeRaI for every projection e; a2R, is calledprojection *-reflexive. A *-ringR is called projection *-reflexiveif0is a projection *-reflexive ideal.
The idealI of the previous definition can not be one-sided ideal, because ifI is a right ideal thenaR1I for everya2I implies1RaI, since1is a projection.
Moreover, the idealI in the definition need not be self-adjoint; indeed, for a field F the *-ring FL
F with the exchange involution, possesses the non self-adjoint projection *-reflexive ideal.0; F /.
It is evident from the definition that *-reflexive and idempotent reflexive *-rings are projection *-reflexive. Accordingly, we raise the following two questions.
Is there a projection *-reflexive *-ring which is not idempotent reflexive?
Is there a projection *-reflexive *-ring which is not *-reflexive?
The answers of these questions are in the following example.
Example 12. The *-ring RD
F F 0 F
over a field F with the involution * defined by
a b 0 c
D
c b 0 a
, is projection *-reflexive because
1 0 0 1
and
0 0 0 0
are the only projections ofR. Clearly,Ris not idempotent reflexive, since the idempotent
1 0 0 0
ofRsatisfies 0 2
0 0
F F 0 F
1 0 0 0
D
0 0 0 0
while
1 0 0 0
F F 0 F
0 2 0 0
D
0 F 0 0
¤0:
Moreover,Ris not *-reflexive, since 0 1
0 1
F F 0 F
0 1 0 0
D 0 1
0 1
F F 0 F
0 1 0 0
D
0 0 0 0
while
0 1 0 0
F F 0 F
0 1 0 1
D
0 F 0 0
The proof of the following proposition, which gives an equivalent definition for projection *-reflexive *-rings, is straightforward.
Proposition 18. A *-ringRis projection *-reflexive if and only if for any nonempty subsetAand any projectioneofR,AReD0implieseRAD0.
Obviously, every *-Abelian *-ring is projection *-reflexive and consequently every
*-ring having quasi-*-IFP is also projection *-reflexive, by Proposition2. However, the converse of this statement needs additional condition, as in the next proposition.
Proposition 19. A *-ringRis *-Abelian if and only ifRis projection *-reflexive and satisfieseR.1 e/ReD0for every projectioneofR.
Proof. The necessity is obvious, For sufficiency, letebe an arbitrary projection of the projection *-reflexive *-ringRandeR.1 e/ReD0. By Proposition 18, we have eReR.1 e/D0and taking involution gives.1 e/ReReD0. Hence,.1 e/ReD0 which implies thate is semicentral, from [Lemma 1.1, [8]], and hence it is central.
ThusRis *-Abelian
In the next result we show when a projection in a projection *-reflexive *-ring is central.
Proposition 20. LetR be a projection *-reflexive *-ring ande is a projection of R. Then the following are equivalent:
(i) eis central.
(ii) eRis a projection-*-reflexive *-ideal.
Proof. .i /).i i /: Assume thataRf eRfor some projectionf ofR. So that arf Dearf for everyr2R and hence.1 e/aRf D0. Therefore f R.1 e/aD 0D.1 e/f Ra, sinceRis projection *-reflexive, and consequentlyf RaDef Ra eR. HenceeRis a projection-*-reflexive ideal.
.i i /).i /: is clear sinceeRis a *-ideal and soeis central.
Corollary 4. If every principal *-ideal of R is projection *-reflexive, then R is
*-Abelian.
Finally, Since the only projections of the *-cornereRe is the projection e, then eReis projection *-reflexive ifRis projection *-reflexive.
Proposition 21. LetR be a projection *-reflexive *-ring, then the *-cornereRe, for every projectioneofR, is also projection *-reflexive.
6. *-NILPOTENCY IN *-REVERSIBLE*-RINGS
According to [3], in a *-ringRevery *-nilpotent element is nilpotent but the con- verse is not always true as shown in [3, Example 2.2]. In the next, we give a sufficient condition that makes a nilpotent element *-nilpotent.
Proposition 22. In a *-reversible *-ringR, every nilpotent element is *-nilpotent.
Proof. Letabe a nilpotent element of a *-reversible *-ringR. HenceanD0, for some positive integern, and multiplying by aform right, we get an 1.aa/D0.
From the *-reversibility ofR, we have.aa/an 1D0. Multiply again byaform right and apply the *-reversible property, we get .aa/an 2D0. Continuing this
process, we get.aa/nD0andais *-nilpotent.
However, the *-reversibility condition in the previous proposition is sufficient but not necessary as clear from Example 6. Indeed, the elements of the *-ideal 0
@
0 D D 0 0 D 0 0 0
1
Aare precisely all the nilpotent (which also *-nilpotent) elements of the ringR.
Corollary 5. Every *-reduced *-reversible *-ring is reduced.
By the definition of nilpotency, an element is nilpotent if and only if a power of it is also nilpotent. This is not the case for *-nilpotent elements as shown in the following examples.
Example 13. In the *-ring RDM2.C/ of 22 matrices with complex entries and transpose involution, the elementaD
p3C{
2 1
1
p3 { 2
!
satisfies.aa/6D 1 1
1 1
which can not tend to zero ever with any power. Thus a is not - nilpotent, while.a3.a3//1D.a3/2D0which means thata3is *-nilpotent.
In the next, a sufficient condition is given to make *-nilpotency transfers between the element and its powers.
Lemma 2. In a *- reversible *-ringR, the elementais *-nilpotent if and only if a2is *-nilpotent.
Proof. Letabe a *-nilpotent element ofR, thenanD.aa/mD0, for some posit- ive integersmandn. Now,0D.aa/mDa.aa/mDa.aa/m 1.aa/and from the *-reversibility ofR, we get0D.aa/a.aa/m 1Da.a/2.aa/m 1. Multiply the last equation byafrom right to geta.a/2a.aa/m 2.aa/D0and applying the
*-reversible property again, we get aa2.a/2a.aa/m 2 D 0 D aa2.a/2.aa/m 2a. Multiply again byafrom right and apply the *-reversibility, we geta.a/2a2.a/2.aa/m 2D0. Continuing, we get.a2.a/2/mD0anda2is
*-nilpotent.
For sufficiency, if a2 is *-nilpotent; that is .a2/n D0D.a2.a/2/m for some positive integersmandn, we get by the same procedure as above.aa/4mD0and
ais *-nilpotent.
Proposition 23. In a *- reversible *-ringR, the elementa is *-nilpotent if and only ifakis also *-nilpotnet for every positive integerk.
Proof. The sufficient condition is clear. For the necessity, letabe a *-nilpotent ele- ment ofR, thenalD.aa/nD0for some positive integerslandn. We use induction onkto show thatak.a/kis nilpotent. The casekD2is clear from Lemma2. Now, we have to show thatakC1.a/kC1is also nilpotent ifak.a/k is nilpotent. Now, if 0D.ak.a/k/mDak.a/k.ak.a/k/m 1, multiply by.a/kC1afrom left and ap- ply the *-reversibility, we get.a/k.ak.a/k/m 1.a/kC1akC1D0. Multiply bya
from left and take involution of both sides, we obtain
.a/kC1akC1.ak.a/k/m 1akC1 D 0. The *-reversibility of R gives ak.a/k.ak.a/k/m 2akC1.a/kC1akC1D0. Multiplying by .a/kC1afrom left gives.a/kC1akC1.a/k.ak.a/k/m 2akC1.a/kC1akC1D0and the *-reversibi- lity of R gives.a/k.ak.a/k/m 2akC1..a/kC1akC1/2D0. Multiply again by .a/kC1, we get .a/k.ak.a/k/m 2.akC1.a/kC1/3 D0. Continuing, we get .a/k.akC1.a/kC1/2m 1 D 0 and multiplication by akC1a gives
.akC1.a/kC1/2mD0.
Conjecture 1(K¨othe’s conjecture). If a ring has a non-zero nil right ideal, then it has a nonzero nil ideal, is still unsolved.
In [9, Theorem 2.2], Cohn proved that for reversible rings, K¨othe’s conjecture has an affirmative solution. In the next, we have a strong affirmative solution for
*-reversible *-rings.
Proposition 24. Every *-reversible *-ring which is not *-reduced, contains a nonzero nilpotent ideal.
Proof. IfRis not *-reduced and *-reversible *-ring, thenRcontains a nonzero *- nilpotent element, saya. So thatamD.aa/nD0, for some positive integersmand n. IfnD1, we haveamDaaD0which impliesr1amDr1am 1aD0for every r12R. From the *-reversibility ofR, we getar1am 1D0. Againr2ar1am 1D r2ar1am 2aD0impliesar2ar1am 2D0for everyr1; r22R. Continuing, we get .RaR/mD0; that is the ideal generated byais a nonzero nilpotent ideal. Ifn > 1, we haveaa¤0. Since.aa/nD0, thenr1.aa/nD0gives.aa/r1.aa/n 1D0 due to the self-adjointness of aa and using the *-reversible property. As before, we get .RaaR/nD0; that is the *-ideal generated byaa is a nonzero nilpotent
ideal.
Corollary 6. In a *-reversible *-ringR, ifRhas a non-zero nil right ideal, then it has a nonzero nil ideal.
Corollary 7. Each semiprime *-reversible *-ring is *-reduced.
CONCLUSION
We can now sate the following implications in the class of rings with involution.
*-reducedks proper +3semiproper
Abelian
domain +3reduced +3
KS
reversible +3
reflexive +3
+3idempotent reflexive *-domain +3*-reversible +3
*-reflexive +3projection *-reflexive
*-IFP +3IFP +3quasi-*-IFP +3*-Abelian
KS
REFERENCES
[1] U. A. Aburawash and M. Saad, “On biregular, IFP and quasi-Baer-rings,”East-West J. Math., vol. 16, no. 2, pp. 182–192, 2014.
[2] U. A. Aburawash and K. B. Sola, “-zero divisors and-prime ideals,”East-West J. Math., vol. 12, no. 1, pp. 27–31, 2010.
[3] U. A. Aburawash and M. Saad, “-Baer property for rings with involution,”Studia Sci. Math.
Hungar, vol. 53, no. 2, pp. 243–255, 2016, doi:10.1556/012.2016.53.2.1338.
[4] K. I. Beidar, L. M´arki, R. Mlitz, and R. Wiegandt, “Primitive involution rings,”Acta Mathematica Hungarica, vol. 109, no. 4, pp. 357–368, 2005, doi:10.1007/S10474-005-0253-4.
[5] H. E. Bell, “Near-rings in which each element is a power of itself,”Bull. Austral. Math. Soc., vol. 2, pp. 363–368, 1970, doi:10.1017/S0004972700042052.
[6] G. F. Birkenmeier and N. J. Groenewald, “Prime ideals in rings with involution,”Quaest. Math., vol. 20, no. 4, pp. 591–603, 1997, doi:10.1080/16073606.1997.9632228.
[7] G. F. Birkenmeier, “Idempotents and completely semiprime ideals,”Comm. Algebra, vol. 11, pp.
567–58, 1983, doi:10.1080/00927878308822865.
[8] G. F. Birkenmeier, J. Y. Kim, and J. K. Park, “Quasi-Baer ring extensions and biregular rings,”
Bull. Austral. Math. Soc., vol. 61, no. 1, pp. 39–52, 2000, doi:10.1017/S0004972700022000.
[9] P. M. Cohn, “Reversible rings,” London Math. Soc., vol. 31, pp. 641–648, 1999, doi:
10.1112/S0024609399006116.
[10] J. Y. Kim and J. U. Baik, “On idempotent reflexive rings,”Kyungpook Math. J., vol. 46, pp. 597–
601, 2006.
[11] N. K. Kima and Y. Lee, “Extensions of reversible rings,” J. Pure App. Algebra, vol. 185, pp.
207–223, 2003, doi:10.1016/S0022-4049(03)00109-9.
[12] T. K. Kwak and Y. Lee, “Reflexive property of rings,”Comm. Algebra, vol. 40, no. 4, pp. 1576–
1594, 2012, doi:10.1080/00927872.2011.554474.
[13] G. Mason, “Reflexive ideals,” Comm. Algebra, vol. 9, pp. 1709–1724, 1981, doi:
10.1080/00927878108822678.
Authors’ addresses
Usama A. Aburawash
Alexandria University, Department of Mathematics and Computer Science, Alexandria, Egypt E-mail address:aburawsh@alexu.edu.eg
Muhammad Saad
Alexandria University, Department of Mathematics and Computer Science, Alexandria, Egypt E-mail address:muhammad.saad@alex-sci.edu.eg
