Advances in Pure Mathematics, 2012, 2, 217-219 Published Online July 2012 (
On Commutativity of Semiprime Right Goldie Ck-Rings
Najat M. Muthana, S. Khalid Nauman
Department of Mathematics, King Abdulaziz University, Jeddah, KSA
Email: {najat_muthana, synakhaled}
Received December 2, 2011; revised February 5, 2012; accepted February 14, 2012
This short exposition is about some commutativity conditions on a semiprime right Goldie Ck-ring. In particular, it is
observed here that a semiprime right Goldie Ck-ring with symmetric quotient is commutative. The statement holds if the
symmetric ring is replaced by reduced, 2-primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring.
Keywords: Semiprime Right Goldie Ck-Rings; Reduced; Symmetric; Von Neumann Regu lar Rings
In this short note we expose some commutativity condi-
tions on a semiprime right Goldie Ck-ring. All rings here
are associative with an identity. A ring A is said to be a
Ck-ring, as introduced by Chuang and Lin in 1989 in [1],
if for very pair of elements x, y
A, there exist integers
m = m(x, y) and n = n(x, y) such that
where [x, y]k is the kth-commutator defined by Klein,
Nada, and Bell in [2] in 1980, as
A ring is called a symmetric ring (in the sense of Lam-
bek [3]), if when ever rab = 0, then rba = 0, ,
semiprime (respt. reduced) if A has no non-zero nilpotent
ideal (respt. element) and von Neumann regular if for
each a
A, there exists r
A such that ara = a. A ring
is right Goldie in case it has finite right uniform dimen-
sion and satisfies acc on right annihilators.
In [1; Theorem 1] Chaung-Lin proved that:
Lemma 1: Every reduced Ck-ring is commutative.
We use it to prove the following.
Theorem: A semiprime right Goldie Ck-ring with sym-
metric right quotient is commutative.
Proof: Lambek in [3; Section 1G] proved that every
reduced ring is symmetric. We prove that the converse
holds for von Neumann regular rings. In deed, one may
deduce easily that A is symmetric if and only if
12 0
aa a
  
12 0
pp pk
aaa ,
where ai, ap(i)
A and p is a one-to-one correspondence
on the set {1, 2, ···, k}. Let a
N(A) be a non-zero ele-
ment of some index n. Since A is von Neumann regular,
for some x
aaxa axa
 .
But A is symmetric and an = 0, which implies that a =
anxn1 = 0. Hence A is reduced.
The famous Goldie’s Theorem states that a ring A is
semiprime right Goldie if and only if A has a right quo-
tient ring B which is semisimple Artinian [4; Theorem
2.3.6]. But a semisimple Artinian ring is von Neumann
regular [5; Theorem 1.7]. Since B is symmetric and now
von Neumann regular, therefore B is reduced. This
means that A is reduced. Since A is a C
k-ring, by the
Lemma 1, we get that A is commutative.
The statement of the Theorem remains unchanged if
we replace the condition of the ring being symmetric by
2-primal, abelian, left or right duo, NI, NCI, IFP, quasi-
IFP, near-IFP, Armendariz, weak-Armendariz, and some
other relations that are listed in Lemma 2.
Let us denote by N(A) the set of all nilpotent elements
of A. For a reduced ri ng N(A) = 0, and a ring is NI if N(A)
is an ideal [6], NCI if N(A) contains a nonzero ideal [7],
and 2-primal if N(A) is the intersection of prime ideals
[8]. A ring A is said to have “Insertion of factor property
(in short, IFP) [9] in case for any pair of elements a, b of
A, if ab = 0, then arb = 0 for all . Such rings are
also termed as semicommutative in literature, we simply
call them IFP rings. Near-IFP (respt. quasi-IFP) rings
are introduced recently in [10] (respt. in [11]), and are
characterized asAaA contains a non-zero nilpotent ideal
of A for any
in [10; Proposition 1.2] (respt.
AaA is a nilpotent ideal of A for any in [11;
Lemma 1.3]). 0aA
By definitions, every reduced ring is an IFP ring, an
IFP ring is a quasi-IFP ring, and a quasi-IFP ring is a
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near-IFP ring. The converse need not be true in general
(see the Example below) but for a semiprime ring it
A ring A is called Armendariz in [12] if whenever
polynomials in
01 m
xaax ax 
xbb bxx
 
0fxgx satisfy
, then
for each i, j, and weak Armendariz in [13] if
whenever , then for each i, j,
01 0bbx
For several interactions and various characterizations
with examples and counter examples of these rings which
we have discussed above the interested reader may refer
to the articles [4,11,12,14,15].
Example: It is clear from the examples and counter
examples in above citations that the rings listed above are
different from each other, but we found no example for
near-IFP and quasi-IFP rings to be different in literature.
By definitions, quasi-IFP is near-IFP, we prove that the
opposite may not be true.
Let R be a ring and a nilpotent ideal of R such
that every element of is a unit. For example, a
local ring is of this type. By Proposition 1.10 [10]
is near-IFP. Let. It is
clear that
0ij n
is nilpotent if , otherwise not nilpotent in general.
The ideal n
of n is proper and nilpotent.
Hence we conclude that is not quasi-IFP.
Now we give a list of rings which coincide on the
condition of von Neumann regularity. Here P stands for
some property, for instance, property for being reduced,
Lemma 2: Let A be a von Neumann regular ring. Then
the following are equivalent.
(P1) A is reduced;
(P2) A is left (or right) duo;
(P3) A is abelian;
(P4) A is 2-primal;
(P5) A is symmetric;
(P6) A is NI;
(P7) A is NCI;
(P8) A is IFP;
(P9) A is quasi-IFP;
(P10) A is near-IFP;
(P11) A is a subdirect product of division ring;
(P12) A is Armendariz;
(P13) A is weak Armendariz;
(P14) If such that with
, then
,, ,aaaA
0.aa a
0aa a
 
(P15) If such that , then
,, ,aaaA
0aa a
 
0.aa a
Proof: The equivalence (P1) (P5) is proved in the
Theorem above. Equivalences of (P1)-(P4) and (P6) and
(P7) hold in [11; Proposition 1.4]. It is clear by defini-
tions that every reduced ring is IFP, every IFP ring is
quasi-IFP and every quasi-IFP ring is near IFP. Thus
(P1) (P8) (P9) (P10).
Because a von Neumann regular ring is semiprime, by
[10; Proposition 1.4], for a semiprime ring a near-IFP
ring is reduced, giving the equivalence (P1)
Equivalences of (P1)-(P3), (P6), (P8) and (P10) also hold
in [10; Proposition 1.6]. The equivalence of a von Neu-
mann regular ring to be reduced, Armendariz, and weak
Armendariz is proved in [14; Lemma 2.4]. Finally, the
(P8) (P11)
are established in [14; Lemma 2.4].
Lemma 3 Let A be a semiprime ring of bounded index
of nilpotency. Then the following conditions are equiva-
(P1) A is reduced;
(P4) A is 2-primal;
(P6) A is NI;
(P7) A is NCI;
(P8) A is IFP;
(P9) A is quasi-IFP.
Proof: (P1)
(P6) (P7) hold by [7;
Proposition 1.3], (P1)
(P6) (P8) (P10) hold
by [10; Proposition 1.5] while (P1) (P4)
(P9) hold in [11; Proposition 1.6].
The consequences of the Theorem and above lemmas
are the following.
Corollary 1: A Ck-von Neumann regular ring is com-
mutative if any one of the properties (P1)-(P15) of Lemma
2 is satisfied.
Corolla ry 2: A Ck-semiprime ring of bounded index of
nilpotency is commutative if any one of the properties
(P1), (P4), (P6)-(P9) of Lemma 3 is satisfied.
Corollary 3: Let A be a semiprime right Goldie ring
and B its classical ring of quotients. Then the ring B sat-
isfies all equivalent conditions from (P1) to (P15) of
Lemma 2. Moreover, for the ring A, the conditions (P1),
(P4), (P8), (P9), (P10), (P12) and (P13) of Lemma 2 are
mutually equivalent and are also equivalent to above
each of fifteen conditions for the ring B.
Proof: Equivalence of (P1) to (P15) is followed from
[14; Theorem 2.6] and Lemma 1.2.
If A is near-IFP and semiprime, and if a is nilpotent,
then every ideal of AaA is zero. Hence, in particular, a is
zero, and A is reduced. So, (P1), (P8)-(P10) are equiva-
lent for the ring A. Equivalence of (P1) and (P4) for the
ring A is obvious and for the same ring (P12) and (P13)
are followed from [14; Theorem 2.6].
Corollary 4: A semiprime right Goldie Ck-ring is
commutative if its classical ring of quotient satisfies any
Copyright © 2012 SciRes. APM
Copyright © 2012 SciRes. APM
one of the properties (P1)-(P15) as listed in Lemma 2.
This project was funded by the Deanship of Scientific
Research (DSR), King Abdulaziz University, Jeddah,
under grant no. 3-043/429. The authors, therefore, ac-
knowledge with thanks DSR technical and financial
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