Generalized Semicommutative and Skew Armendariz Ideals
We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts.
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| author | Nikmehr, M. J. Нікмер, М. Дж. |
| author_facet | Nikmehr, M. J. Нікмер, М. Дж. |
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| description | We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts. |
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UDC 512.5
M. J. Nikmehr (K. N. Toosi Univ. Technology, Tehran, Iran)
GENERALIZED SEMICOMMUTATIVE AND SKEW ARMENDARIZ IDEALS
УЗАГАЛЬНЕНI НАПIВКОМУТАТИВНI ТА КОСI IДЕАЛИ АРМЕНДАРIЗА
We generalize the concepts of semicommutative, skew Armendariz, Аbelian, reduced, and symmetric left ideals and study
the relations between them.
Узагальнено поняття напiвкомутативних косих абелевих зведених та симетричних лiвих iдеалiв Армендарiза та
вивчено спiввiдношення мiж ними.
1. Introduction. Throughout this paper R denotes an associative ring with identity 1 and α denotes
a nonzero and nonidentity endomorphism of a given ring with α(1) = 1, and 1 denotes identity
endomorphism, unless specified otherwise.
We write R[x], for the polynomial ring, moreover, R[x, α] =
{∑n
i=0
aix
i
∣∣∣ n ≥ 0, ai ∈ R
}
becomes a ring under the following operation:
f(x) =
n∑
i=0
aix
i, g(x) =
m∑
j=0
bjx
j ∈ R[x, α], f(x)g(x) =
m+n∑
k=0
( ∑
i+j=k
aiα
i(bj)
)
xk.
The ring R[x, α] is called the skew polynomial extension of R.
In [4], Baer-rings are introduced as rings in which the right (left) annihilator of every nonempty
subset is generated by an idempotent. According to Clark [5], a ring R is said to be quasi-Baer ring
if the right annihilator of every right ideal of R is generated (as a right ideal) by an idempotent. A
ring R is called right principally quasi-Baer ring if the right annihilator of a principally right ideal
of R is generated (as a right ideal) by an idempotent. Finally, a ring R is called right principally
projective ring if the right annihilator of an element of R is generated by an idempotent [4].
For an endomorphism α of ring R, Hong, Kim, and Kowak [7] called R an α-skew Armendariz
ring if whenever polynomials f(x) =
∑n
i=0
aix
i and g(x) =
∑m
j=0
bjx
j ∈ R[x, α], f(x)g(x) = 0
then aiαibj = 0 for each i and j. Some properties of Armendariz rings have been studied in [9 – 11].
In [2], the notions of α-Abelian, α-semicommutative, α-reduced, α-symmetric and α-Armendariz
rings have been introduced which generalize Abelian, semicommutative, reduced, symmetric and
Armendariz rings. Aghayev et al. defined a ring R is called α-Abelian if, for any a, b ∈ R, and
any idempotent e ∈ r, ea = ae and ab = 0 if and only if aα(b) = 0 and a ring R is called α-
semicommutative if, for any a, b ∈ R, ab = 0 implies aRb = 0 and ab = 0 if and only if aα(b) = 0.
A ring R is called α-reduced if, for any a, b ∈ R, ab = 0 implies aR ∩ Rb = 0 and ab = 0 if and
only if aα(b) = 0. A ring R is called α-symmetric if, for any a, b, c ∈ R, abc = 0 implies acb = 0
and ab = 0 if and only if aα(b) = 0.
They proved that α-semicommutative, α-reduced, α-symmetric and α-Armendariz rings are α-
Abelian. For a right principally projective ring R, they also proved the following conditions on
α-reduced of a ring R are equivalent:
c© M. J. NIKMEHR, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9 1213
1214 M. J. NIKMEHR
α-symmetric ⇔ α-semicommutative
m m
α-Abelian ⇔ α-Armendariz
.
In this paper we introduce the concepts of α-Abelian, α-semicommutative, α-reduced, α-symmetric
and α-skew Armendariz left ideals and investigate their properties. Moreover, we prove that if
there exists a classical right quotient ring Q of a ring R consisting of central elements and I is
α-semicommutative left ideal of R, then QI is α-semicommutative left ideal of Q(R).
Similarly we prove that if I is a left ideal of a ring R and ∆ is a multiplicatively closed subset
of R consisting of central elements and I is α-semicommutative left ideal of R, then ∆−1I is
α-semicommutative left ideal of ∆−1R.
2. Semicommutative and skew Armendariz ideals. In this section the notion of an α-Abelian
left ideals is introduced as a generalization of Abelian left ideals. We recall that a left ideal I of R
is called Abelian if for any a, b ∈ R and any idempotent e ∈ R, ea− ae ∈ rR(I). Now we have the
following definition.
Definition 2.1. A left ideal I of R is called α-Abelian if, for any a, b ∈ R and any idempotent
e ∈ R, we have the following conditions:
1) ea− ae ∈ rR(I),
2) ab ∈ rR(I) if and only if aα(b) ∈ rR(I).
So a left ideal I is Abelian if and only if it is 1-Abelian. The following example shows that there
exists an Abelian left ideal, but it is not α-Abelian left ideal.
Example 2.1. Let R be the ring Z
⊕
Z with the usual componentwise operation. It is clear that
R is an Abelian ring. Let α : R → R be defined by α((a, b)) = (b, a). Then (1, 0)(0, 1) = 0, but
(1, 0)α((0, 1)) 6= 0. Hence R is not α-Abelian. If ideal I = R then rR(I) = 0 and then I is an
Abelian left ideal, but it is not an α-Abelian left ideal.
Definition 2.2. A left ideal I of R is called semicommutative if, for any a, b ∈ R, ab ∈ rR(I)
then aRb ⊆ rR(I).
Definition 2.3. A left ideal I of R is called α-semicommutative if, for any a, b ∈ R we have the
following conditions:
1) ab ∈ rR(I) then aRb ⊆ rR(I),
2) ab ∈ rR(I) if and only if aα(b) ∈ rR(I).
So a left ideal I is semicommutative if and only if it is 1-semicommutative.
In general the reverse implication in the above definition does not hold by the following example
which also shows that there exist an endomorphism α of a ring R and left ideal I of R such that I is
semicommutative but is not α-semicommutative.
Example 2.2. Let Z2 be the ring of integers modulo 2 and consider a ring R = Z2
⊕
Z2 with
the usual addition and multiplication. If I = Z2
⊕
0 be a left ideal of R then rR(I) = 0
⊕
Z2.
Now, let α : R → R be defined by α((a, b)) = (b, a). Then α is an automorphism of R. It is clear
that I is semicommutative left ideal. For a = (1, 0) and b = (0, 1) ∈ R, ab = (0, 0) ∈ rR(I) but
aα(b) = (1, 0) /∈ rR(I).
Lemma 2.1. If the left ideal I of R is α-semicommutative, then I is α-Abelian.
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GENERALIZED SEMICOMMUTATIVE AND SKEW ARMENDARIZ IDEALS 1215
Proof. If e is an idempotent in R, then e(1− e) = 0 ∈ rR(I). Since I is α-semicommutative,
we have eα(1 − e) = 0 ∈ rR(I) for any a ∈ R and so ea − eae ∈ rR(I). Similarly, (1 − e)e =
= 0 ∈ rR(I). Since I is α-semicommutative (1−e)ae = 0 ∈ rR(I). So ae−eae ∈ rR(I). Therefore,
ae− ea ∈ rR(I). Thus I is α-Abelian.
Lemma 2.1 is proved.
The following example shows that the condition α(1) = 1 in Lemma 2.1 is not superfluous.
Example 2.3. Let Z be the ring of integers. Consider a ring
R =
{(
a b
0 c
) ∣∣∣∣∣ a, b, c ∈ Z
}
.
If I =
{(
0 b
0 0
) ∣∣∣∣ b ∈ Z
}
be an right ideal of R then lR(I) =
{(
0 b
0 c
) ∣∣∣∣ b, c ∈ Z
}
. Let α :
R → R be defined by α
((
a b
0 c
))
=
(
a 0
0 0
)
. For A =
(
a b
0 c
)
and B =
(
a′ b′
0 c′
)
∈ R, if
AB ∈ lR(I) then we obtain aa′ = 0, and so a = 0 or a′ = 0. This implies ARα(B) ⊆ lR(I) and
thus I is α-semicommutative. Note that α(1) 6= 1 and I is not Abelian.
Definition 2.4. A left ideal I of R is called α-skew Armendariz if the following conditions are
satisfied:
1) for any f(x) =
∑n
i=0
aix
i and g(x) =
∑m
j=0
bjx
j ∈ R[x, α], f(x)g(x) ∈ rR[x,α](I[x])
implies aiαi(bj) ∈ rR(I),
2) ab ∈ rR(I) if and only if aα(b) ∈ rR(I).
We introduce an α-skew Armendariz left ideal in the following example.
Example 2.4. Let R be an α-skew Armendariz ring and consider
S =
{(
a b
c d
) ∣∣∣∣∣ a, b ∈ R
}
.
It is clear that I =
{(
0 b
0 0
)∣∣∣∣ b ∈ R} is the left ideal of S. Let f(x) = A0 + A1x + . . . + Anx
n
and g(x) = B0 + B1x + . . . + Bmx
m ∈ S[x, α], where Ai =
(
a0i a1i
0 a0i
)
, Bj =
(
b0j b1i
0 b0j
)
for
i = 0, . . . , n, j = 0, . . . ,m such that f(x)g(x) ∈ rR[x,α](I[x]). Let
f(x) =
(
α0(x) α1(x)
0 α0(x)
)
, g(x) =
(
β0(x) β1(x)
0 β0(x)
)
,
α0(x) = a00 + a01x+ . . .+ a0nx
n, β0(x) = b00 + b01x+ . . .+ b0mx
m.
Since f(x)g(x) ∈ rR[x,α](I[x]) thus for any h(x) =
(
0 γ(x)
0 0
)
∈ I[x], that γ(x) = γ0 + γ1x+ . . .
. . .+ γtx
t, γ(x)f(x)g(x) = 0. Thus γ(x)α0(x)β0(x) = 0. Since I is α-skew Armendariz left ideal
hence γkαk(a0iα
i(b0j)) = 0 for all k = 0, . . . , t, i = 0, . . . , n and j = 0, . . . ,m. If set k = 0, then
γ0(a0iα
i(b0j)) = 0. Since γ0 ∈ R is arbitrary, thus
(
0 γ0
0 0
)
∈ I. Therefore a0iα
i(b0j) ∈ rR(I),
and hence Aiαi(Bj) ∈ rR(I). Now we consider
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1216 M. J. NIKMEHR(
a1 b1
0 a1
)
,
(
a2 b2
0 a2
)
∈ S,
(
0 c
0 0
)
∈ I,
(
a1 b1
0 a1
)(
a2 b2
0 a2
)
∈ rR(I).
Thus
(
0 c
0 0
)(
a1a2 a1b2 + a2b1
0 a1a2
)
=
(
0 0
0 0
)
, and hence ca1a2 = 0. Since I is α-skew Armen-
dariz, ca1α(a2) = 0. Thus
(
a1 b1
0 a1
)(
a2 b2
0 a2
)
∈ rR(I) if and only if
(
a1 b1
0 a1
)
α
((
a2 b2
0 a2
))
∈ rR(I).
Therefore I is an α-skew Armendariz left ideal.
Proposition 2.1. If I is an α-skew Armendariz left ideal of R and for some a, b, c ∈ R and
some integer n ≥ 1, ab ∈ rR(I) and acnαn(b) ∈ rR(I), then acb ∈ rR(I).
Proof. Consider f(x) = a(1− cx), g(x) = (1 + cx+ . . .+ cn−1xn−1)b ∈ R[x, α], f(x)g(x) =
= ab−acnαn(b)xn ∈ rR[x,α](I[x]). Since I is an α-skew Armendariz left ideal of R, so acb ∈ rR(I).
Proposition 2.1 is proved.
Next, we show that every α-skew Armendariz left ideal of R is an α-Abelian left ideal.
Proposition 2.2. If I is an α-skew Armendariz left ideal of R, then I is an α-Abelian left ideal.
Proof. Assume that I is an α-skew Armendariz left ideal of R. Consider e = e2 ∈ R and
let a = e, b = (1 − e), c = er(1 − e) with r ∈ R. Then clearly ab ∈ rR(I) and c2 = 0 ∈ rR(I)
and hence ac2α2(b) ∈ rR(I) and then by Proposition 2.1, acb ∈ rR(I). So er − ere ∈ rR(I). Let
a1 = 1 − e, b1 = e and c1 = (1 − e)re, we also have a1b1c1 ∈ rR(I). So re − ere ∈ rR(I). Then
re− er ∈ rR(I).
Proposition 2.2 is proved.
Theorem 2.1. Let R be a ring and I, J be left ideals of R. If I ⊆ J and J/I is an α-skew
Armendariz left ideal of R/I, then J is an α-skew Armendariz left ideal of R.
Proof. Let f(x) =
∑n
i=0
aix
i and g(x) =
∑m
j=0
bjx
j ∈ R[x, α] such that f(x)g(x) ∈
∈ rR[x,α](J [x]). Then
∑n
i=0
āix
i
∑m
j=0
b̄jx
j ∈ rR/I[x,α](J/I)[x]). Thus āiαi(b̄j) ∈ rR/I(J/I).
Hence aiαi(bj) ∈ rR(J). Therefore J is an α-skew Armendariz left ideal of R.
Theorem 2.1 is proved.
The following is an immediate corollary of Theorem 2.1.
Corollary 2.1. Let R be a ring and I an left ideal of R. If R/I is α-skew Armendariz then R is
an α-skew Armendariz ring.
A ring R is called locally finite if every finite subset of R generates a finite semigroup multiplica-
tively. Finite rings are clearly locally finite and the algebraic closure of a finite field is locally finite
but it is not finite.
Proposition 2.3. Let R be a locally finite ring and I be an α-skew Armendariz left ideal of R.
Then I is α-semicommutative left ideal of R.
Proof. Let ab ∈ rR(I) with a, b ∈ R. For any r ∈ R, since R is locally finite there exist
integers m, k ≥ 1 such that rm = rm+k. So we obtain inductively rm = rmrk = r2k = . . .
. . . = rmrmk = rm(k+1), put h = k + 1 then rm = rmh with h ≥ 2. Notice that r(h−1)m =
= r(h−2)mrm = r(h−2)mrmh = r2(h−2)m = (r(h−1)m)2. Hence r(h−1)m is an idempotent and
so by Proposition 2.2, ar(h−1)m − r(h−1)ma ∈ rR(I) and abr(h−1)m − r(h−1)mab ∈ rR(I). Thus
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
GENERALIZED SEMICOMMUTATIVE AND SKEW ARMENDARIZ IDEALS 1217
r(h−1)mab ∈ rR(I). On the other hand by Proposition 2.2, ar(h−1)m − r(h−1)ma ∈ rR(I), so
ar(h−1)mb − r(h−1)mab ∈ rR(I), and hence ar(h−1)mb ∈ rR(I). Since I is an α-skew Armendariz
left ideal of R so ar(h−1)mα(h−1)m(b) ∈ rR(I), and by Proposition 2.1, we imply that arb ∈ rR(I)
for all r ∈ R.
Proposition 2.3 is proved.
Let α be an endomorphism of a ring R and Mn(R) be the (n × n)-matrix over ring R and α :
Mn(R) −→ Mn(R) defined by α(aij) = (α(aij)). Then α is an endomorphism of Mn(R). It is
obvious that, the restriction of α to Dn(R) is an endomorphism of Dn(R), where Dn(R) is the
diagonal (n× n)-matrix ring over R. We also denote α |Dn(R) by α.
Proposition 2.4. Let α be an endomorphism of a ring R. Then Dn(I) is an α-skew Armendariz
left ideal of Dn(R) if I is an α-skew Armendariz left ideal for any n.
Proof. Let f(x) = A0 +A1x+ . . .+Apx
p and g(x) = B0 +B1x+ . . .+Bqx
q ∈ Dn(R)[x, α]
satisfying f(x)g(x) ∈ rDn(R)[x,α](Dn(I)[x]), where
Ai =
a
(i)
11 0 . . . 0
0 a
(i)
22 . . . 0
...
...
. . . 0
0 . . . 0 a
(i)
nn
and Bj =
b
(j)
11 0 . . . 0
0 b
(j)
22 . . . 0
...
...
. . . 0
0 . . . 0 b
(j)
nn
.
Then from f(x)g(x) ∈ rDn(R)[x,α](Dn(I)[x]), it follows that
(
p∑
i=0
a(i)
ss x
i
) q∑
j=0
b(j)ss x
j
∈ rR[x,α](I[x]),
for each 1 ≤ s ≤ n. Since I is an α-skew Armendariz left ideal of R, then a(i)
ss αi(b
(j)
ss ) ∈ rR(I) for
any 1 ≤ i ≤ p and 1 ≤ j ≤ q. Therefore
Aiα
i(Bj) =
a
(i)
11α
i(b
(j)
11 ) 0 . . . 0
0 a
(i)
22α
i(b
(j)
22 ) . . . 0
...
...
. . . 0
0 0 . . . a
(i)
nnαi(b
(j)
nn)
∈ rDn(R)(Dn(I)).
Thus it shows that Dn(I) is an α-skew Armendariz left ideal of Dn(R).
Proposition 2.4 is proved.
Every endomorphism σ of rings R and S can be extended to the endomorphism of rings R[x]
and S[x] defined by
∑m
i=0
aix
i →
∑m
j=0
σ(ai)x
i, which we also denote by σ.
Proposition 2.5. Let σ : R→ S be a ring isomorphism, I1 be an ideal of R and I2 be an ideal
of S with σ(I1) = I2. If I2 is an σασ−1-skew Armendariz left ideal of ring S, then I1 is an α-skew
Armendariz left ideal of R.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1218 M. J. NIKMEHR
Proof. Let f(x) =
∑m
i=0
aix
i and g(x) =
∑m
j=0
bjx
j ∈ R[x, α] such that f(x)g(x) ∈
∈ rR[x,α](I1[x]). We set f1(x) = σ(f(x)) =
∑m
i=0
σ(ai)x
i =
∑m
i=0
a′ix
i and g1(x) = σ(g(x)) =
=
∑m
j=0
σ(bj)x
j =
∑m
j=0
b′jx
j ∈ S[x, σασ−1]. First we shall show f(x)g(x) ∈ rR[x,α](I1[x]) im-
plies that f1(x)g1(x) ∈ rS[x,σασ−1](I2[x]). Let I1[x]f(x)g(x) = 0. From the definition of f1(x) and
g1(x),we have σ(I1[x]f(x)g(x)) = I2[x]f1(x)g1(x) = 0. From the fact that I2 is an σασ−1-skew Ar-
mendariz left ideal of ring S, we have a′i(σασ
−1)ib′j ∈ rS(I2). So that (σασ−1)t = σαtσ−1 we obtain
a′i(σα
iσ−1)b′j ∈ rS(I2). Since σ(ai) = a′i and σ(bj) = b′j and σ(I1) = I2, then σ(I1aiα
i(bj)) = 0.
Since σ is an isomorphism, then aiαi(bj) ∈ rR(I1). Clearly the other condition in definition is hold.
Hence I1 is an α-skew Armendariz left ideal of R.
Proposition 2.5 is proved.
As a result, we shall show that, under certain condition, the left ideals of the subring of upper
triangular skew matrices over a ring R have an α-skew Armendariz structure.
Let Eij = (est : 1 ≤ s, t ≤ n) denotes unit (n × n)-matrices over ring R, in which eij = 1
and est = 0 when s 6= i or t 6= j, 0 ≤ i, j ≤ n for n ≥ 2. If V =
∑n−1
i=1
Ei,i+1, then
Vn(R) = RIn +RV + . . .+RV n−1 is the subring of upper triangular skew matrices.
Corollary 2.2. Suppose that α is an endomorphism of a ring R, θ : Vn(R) → R[x]
(xn)
be a ring
isomorphism, I1 is a left ideal of Vn(R) and I2 is a left ideal of
R[x]
(xn)
. If I2 is an α-skew Armendariz
left ideal of
R[x]
(xn)
and θ(I1) = I2, then I1 is an α-skew Armendariz left ideal of Vn(R).
Proof. Assume that I2 is an α-skew Armendariz left ideal of
R[x]
(xn)
and define
θ : Vn(R)→ R[x]
(xn)
by
θ
(
r0In + r1V + . . .+ rn−1V
n−1
)
= r0 + r1x+ . . .+ rn−1x
n−1 + (xn).
Now we have I1 is a θ−1αθ-skew Armendariz left ideal of Vn(R) and that
θ−1αθ(r0In + r1V + . . .+ rn−1V
n−1) = α(r0In + r1V + . . .+ rn−1V
n−1),
which means that I1 is an α-skew Armendariz left ideal of Vn(R).
Corollary 2.2 is proved.
Recall that a ring is reduced if it has no nonzero nilpotent elements. In [2], α-reduced ring is
introduced. A ring R is α-reduced, if for any a, b ∈ R
1) ab = 0 implies aR ∩Rb = 0,
2) ab = 0 if and only if aα(b) = 0.
In this work we define reduced and α-reduced left ideals.
Definition 2.5. A left ideal I of R is called reduced, if for any a, b ∈ R, ab ∈ rR(I), then
aR ∩Rb ⊆ rR(I).
Definition 2.6. A left ideal I of R is called α-reduced, if for any a, b ∈ R, we have the following
conditions:
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GENERALIZED SEMICOMMUTATIVE AND SKEW ARMENDARIZ IDEALS 1219
1) ab ∈ rR(I) then aR ∩Rb ⊆ rR(I),
2) ab ∈ rR(I) if and only if aα(b) ∈ rR(I).
So the left ideal I is reduced if and only if it is 1-reduced.
Lemma 2.2. If I is an α-reduced left ideal of R, then I is an α-semicommutative.
Proof. Suppose ab ∈ rR(I) for any a, b ∈ R. Since I is an α-reduced left ideal of R then
aR∩Rb ⊆ rR(I). Because aRb ⊆ aR∩Rb, then aRb ⊆ rR(I). Therefore I is an α-semicommutative.
Now by Lemma 2.2 we have the following lemma.
Lemma 2.3. If I is an α-reduced left ideal of R, then I is α-Abelian.
Proposition 2.6. Let α be an endomorphism of a ring R and I be an α-reduced left ideal of R.
Then I is an α-skew Armendarize left ideal.
Proof. Let f(x) =
∑n
i=0
aix
i and g(x) =
∑m
j=0
bjx
j ∈ R[x, α] such that f(x)g(x) ∈
∈ rR[x,α](I[x]). Then for each h ∈ I, h
(∑
i+j=l
aiα
i(bj)
)
= 0. Thus
∑
i+j=l
aiα
i(bj) ∈ rR(I)
for l = 0, . . . ,m + n. So ha0b0 = 0. Thus ha0b1b0 = 0, since I is α-semicommutative and
h(a1α(b0) + a0b1) = 0. Multiplying by b0 on the right we have h(a1α(b0) + a0b1)b0 = 0. So we
have h(a1α(b0)b0) = 0. Thus h(a1α
2(b0)) = 0, and then h(a1α(b0)) = 0, since I is α-reduced.
Thus ha0b1 = 0. Assume that s ≥ 1 and h(aiα
i(bj)) = 0 for all i and j with i + j ≤ s. Note
that h(a0bs+1 + a1α(bs) + . . . + as+1α
s+1(b0)) = 0, where ai and bj are 0 if i > n and j > m.
Multiplying by αs(b0) on the right yields
h(a0bs+1α
s(b0) + a1α(bs)α
s(b0) + . . .+ as+1α
s+1(b0)αs(b0)) = 0.
Since I is α-semicommutative and h(aiα
i(b0)) = 0 for i ≤ s, it follows that h(aiRα
i(b0)) = 0.
Thus h(as+1α
s+1(b0)αs(b0)) = h(as+1α(αs(b0))αs(b0)) = 0, which implies h(as+1α
s+1(b0)) = 0
by assumption. So
h(a0bs+1 + a1α(bs) + . . .+ asα
s(b1)) = 0.
Analogously, multiplying by αs−1(b1) on the right yields
h(a0bs+1α
s−1(b1) + a1α(bs)α
s−1(b1) + . . .+ asα
s(b1)αs−1(b1)) = 0.
The similar argument as the above reveals that h(asα
s(b1)αs−1(b1)) = 0. Thus h(asα
s(b1)) = 0.
Continuing this process, we have hasαs(b1) = . . . = ha1α(bs) = ha0bs+1 = 0. So we prove that
haiα
i(bj) = 0 for all i and j with i+ j ≤ s+ 1. By the induction principle, haiαi(bj) = 0 for every
i and j.
Proposition 2.6 is proved.
Definition 2.7. A left ideal I of R is called symmetric, if for any a, b, c ∈ R, abc ∈ rR(I), then
acb ∈ rR(I).
Definition 2.8. A left ideal I of R is called α-symmetric, if for any a, b, c ∈ R,
1) abc ∈ rR(I) then acb ∈ rR(I),
2) ab ∈ rR(I) if and only if aα(b) ∈ rR(I).
So the left ideal I is symmetric if and only if it is 1-symmetric.
Proposition 2.7. If I is an α-symmetric left ideal of R, then I is an α-semicommutative.
Proof. Suppose ab ∈ rR(I), for any a, b ∈ R. Thus abr ∈ rR(I), for any r ∈ R. So arb ∈ rR(I),
since I is α-symmetric. Therefore I is an α-semicommutative.
Now by Proposition 2.7 we have the following corollary.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1220 M. J. NIKMEHR
Corollary 2.3. If I is an α-symmetric left ideal of R, then I is an α-Abelian.
There exists an α-Abelian right ideal which are also α-semicommutative, α-reduced and α-
symmetric.
Example 2.5. Let Z be the ring of integers and Z2×2 the full (2× 2)-matrix ring over Z,
R =
{(
a b
0 d
)
∈ Z2×2
∣∣∣∣∣ a ≡ d(mod2), b ≡ 0(mod2)
}
and
I =
{(
0 b
0 0
)
∈ Z2×2
∣∣∣∣∣ b ≡ 0(mod2)
}
be the right ideal of R. We have
lR(I) =
{(
0 b
0 d
)
∈ Z2×2
∣∣∣∣∣ d ≡ 0(mod2), b ≡ 0(mod2)
}
.
We define α
((
a b
0 d
))
=
(
a 0
0 d
)
. 0, 1 are only idempotents in R and for any A =
(
a b
0 d
)
∈ R
and B =
(
c e
0 h
)
∈ R, AB ∈ lR(I) if and only if ac = 0. Since Z is domain we have a = 0 or
c = 0. If a = 0, then
Aα(B) =
(
0 b
0 d
)
α
((
c e
0 h
))
=
(
0 b
0 d
)(
c 0
0 h
)
=
(
0 0
0 dh
)
∈ lR(I).
If c = 0, then
Aα(B) =
(
a b
0 d
)
α
((
0 e
0 h
))
=
(
a b
0 d
)(
0 0
0 h
)
=
(
0 0
0 dh
)
∈ lR(I).
On the other hand if Aα(B) ∈ lR(I) therefore
(
a b
0 d
)
α
((
c e
0 h
))
∈ lR(I), then
(
ac bh
0 dh
)
∈
∈ lR(I). So ac = 0 and similarly we have AB ∈ lR(I). Therefore I is α-Abelian right ideal of R.
Now we show that I is α-semicommutative right ideal of R. For any A =
(
a b
0 d
)
, B =
(
c e
0 h
)
and C =
(
g k
0 m
)
∈ R, let AB ∈ lR(I) thus ac = 0 and so acg = agc = 0, since a, c, g ∈ Z. We
have
ACB =
(
agc age+ akh+ bhm
0 dmh
)
=
(
0 age+ akh+ bhm
0 dmh
)
∈ lR(I).
I is α-symmetric right ideal of R, since ABC ∈ lR(I) iff acg = 0, iff agc = 0. Therefore ACB ∈
∈ lR(I). Now we show that I is α-reduced right ideal ofR. LetAB ∈ lR(I), then ac = 0. Thus a = 0
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
GENERALIZED SEMICOMMUTATIVE AND SKEW ARMENDARIZ IDEALS 1221
or c = 0. Now if X ∈ AR ∩ RB, then there exist K =
(
k1 k2
0 k3
)
∈ R and G =
(
g1 g2
0 g3
)
∈ R,
such that X = AK = GB. But AK =
(
ak1 ak2 + bk3
0 dk3
)
and GB =
(
g1c g1e+ g2h
0 g3h
)
. Thus
g1c = ak1. If a = 0, then X =
(
0 bk3
0 dk3
)
∈ lR(I) and if c = 0, then X =
(
0 g2h
0 g3h
)
∈ lR(I).
Therefore I is α-reduced right ideal of R.
Recall that rR(
⊕
Ii) =
⋂
rR(Ii). Now we have the next proposition.
Proposition 2.8. For any index set Γ, if Ii is an α-Abelian left ideal of R for each i ∈ Γ, then⊕
i∈Γ Ii is an α-Abelian left ideal of R.
Theorem 2.2. Suppose that I is left ideal a ring R and ∆ is a multiplicatively closed subset of
R consisting of central regular elements. We have the following conditions:
1. If I is an α-semicommutative left ideal of R, then ∆−1I is an α-semicommutative left ideal
of ∆−1R.
2. If I is an α-symmetric left ideal of R, then ∆−1I is an α-symmetric left ideal of ∆−1R.
3. If I is an α-reduced left ideal of R, then ∆−1I is an α-reduced left ideal of ∆−1R.
Proof. We employ the method used in the proof of [8] (Proposition 3.1). For instance, we
prove (1). Let βγ ∈ r∆−1R(∆−1I) with β = u−1a, γ = v−1b, u,v ∈ ∆ and a,b ∈ R. Since
∆ is contained in the center of R, we have 0 = ∆−1Iβγ = ∆−1Iu−1av−1b = ∆−1Iab(uv)−1.
So Iab = 0. It follows that arb ∈ rR(I) for all r ∈ R, since I is an α-semicommutative left
ideal of R. Now for δ = w−1r with w ∈ ∆ and r ∈ R , ∆−1Iβδγ = ∆−1Iarb(uwv)−1 = 0.
Thus βδγ ∈ r∆−1R(∆−1I). Now suppose that βγ ∈ r∆−1R(∆−1I). Therefore 0 = ∆−1Iβγ =
= ∆−1Iu−1av−1b = ∆−1Iab(uv)−1 iff Iab = 0, iff Iaα(b) = 0, iff ∆−1Iaα(b)(uv)−1 = 0, iff
βα(γ) ∈ r∆−1R(∆−1I), since I is an α-semicommutative left ideal of R and α is endomorphism of
R and α(γ) = v−1α(b). Hence ∆−1I is an α-semicommutative left ideal of ∆−1R.
Theorem 2.2 is proved.
A ring of R is called right Ore if given a,b ∈ R with b regular there exist a1, b1 ∈ R with b1
regular such that ab1 = ba1. It is a well-known fact that R is a right Ore ring if and only if there
exists a classical right quotient ring of R.
Theorem 2.3. Suppose that there exists a classical right quotient Q of a ring R consisting of
central elements. We have the following conditions:
1. I is an α-semicommutative left ideal of R if and only if QI is an α-semicommutative left ideal
of Q.
2. I is an α-symmetric left ideal of R if and only if QI is an α-symmetric left ideal of Q.
3. I is an α-left reduced ideal of R if and only if QI is an α-reduced left ideal of Q.
Proof. For instance, we prove (1). Let βγ ∈ rQ(QI) with β = u−1a, γ = v−1b,u, v ∈ R
and a, b ∈ R. Since Q is contained in the center of R, we have 0 = QIβγ = QIu−1av−1b =
= QIab(uv)−1, so Iab = 0. It follows that arb ∈ rR(I) for all r ∈ R, since I is an α-
semicommutative ideal ofR.Now for δ = w−1r withw ∈ R and r ∈ R, QIβδγ = QIarb(uwv)−1 =
= 0. Thus βδγ ∈ rQ(QI). Now suppose that βγ ∈ rQ(QI). Therefore 0 = QIβγ = QIu−1av−1b =
= QIab(uv)−1 iff Iab = 0, iff Iaα(b) = 0, iff QIaα(b)(uv)−1 = 0, iff βα(γ) ∈ rQ(QI), since I
is an α-semicommutative ideal of R and α is endomorphism of R and α(γ) = v−1α(b). Hence QI
is an α-semicommutative left ideal of Q.
Theorem 2.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1222 M. J. NIKMEHR
Let α be an automorphism of a ring R. Suppose that there exists a classical right quotient Q of a
ring R. Then for any b−1a ∈ Q, where a, b ∈ R with b regular the induced map α : Q(R)→ Q(R)
defined by α(b−1a) = (α(b))−1α(a) is also an automorphism.
Proposition 2.9. Suppose that there exists a classical right quotient Q of a ring R consisting of
central elements. If I is α-semicommutative left ideal of R, then I is α-skew Armendariz left ideal
of R if and only if QI is α-skew Armendariz left ideal of Q.
Proof. Suppose that I is α-skew Armendariz. Let f(x) = s−1
0 a0 +s−1
1 a1x+ . . .+s−1
m amx
m and
g(x) = t−1
0 b0+t−1
1 b1x+. . .+ t−1
n bnx
n ∈ QI[x, α] such that f(x)g(x) ∈ rQI[x,α](QI[x]). Let C be a
left denominator set. There exist s, t ∈ C and a′i, b
′
j ∈ R such that s−1
i ai = s−1a′i and t−1
j bj = t−1b′j
for i = 0, 1, . . . ,m and j = 0, 1, . . . , n. Then s−1(a′0+a′1x+. . .+a′mx
m)t−1(b′0+b′1x+. . .+b′nx
n) ∈
∈ rQI[x,α](QI[x]). It follows that (a′0+a′1x+. . .+a′mx
m)t−1(b′0+b′1x+. . .+b′nx
n) ∈ rQI[x,α](QI[x]).
Thus (a′0t
−1 + a′1(α(t))−1x + . . . + a′m(αm(t))−1xm)(b′0 + b′1x + . . . + b′nx
n) ∈ rQI[x,α](QI[x]).
For a′i(α
i(t))−1, i = 0, 1, . . . , n, there exist t′ ∈ C and a′′i ∈ R such that a′i(α
i(t))−1 = t′−1a′′i .
Hence t′−1(a′′0 + a′′1x + . . . + a′′mx
m)(b′0 + b′1x + . . . + b′nx
n) ∈ rQI[x,α](QI[x]). We have (a′′0 +
+ a′′1x + . . . + a′′mx
m)(b′0 + b′1x + . . . + b′nx
n) ∈ rR[x,α](I[x]). Since I is α-skew Armendariz,
so a′′i α
i(b′j) ∈ rR[x,α](I[x]) for all i and j. Since I is α-semicommutative, by Theorem 2.3, QI
is α-semicommutative. Then t′−1a′′i α
i(b′j) ∈ rQ(QI). So a′iα
i(t−1b′j) = (a′i(α
i(t)))−1αi(b′j) =
= ((t′−1a′′i )α
i(b′j)) ∈ rQ(QI). Similarly we have (s−1
i a′i)α
i(t−1
j b′j) = (s−1
i a′i)α
i(t−1b′j) ∈ rQ(QI).
Let βγ ∈ rQ(QI) with β = u−1a, γ = v−1b, u, v ∈ R and a, b ∈ R. Therefore 0 = QIβγ =
= QIu−1av−1b = QIab(uv)−1 iff Iaα(b) = 0, iffQIaα(b)(uv)−1 = 0, iffQIaα(b)u−1(α(v))−1 =
= 0, iff QI(u−1a)((α(v))−1α(b)) = 0, iff QIβα(γ) = 0, since I is α-skew Armendariz and α is
an automorphism of R and Q is contained in the center of R. Thus QI is α-skew Armendariz. The
converse is clear.
Proposition 2.9 is proved.
1. Aghayev N., Gungoroglu G., Harmanci A., Halicioglu S. Abelian modules // Acta Math. Univ. Comenianae. – 2009. –
2. – P. 235 – 244.
2. Aghayev N., Harmanci A., Halicioglu S. On Abelian rings // Turk. J. Math. – 2010. – 34. – P. 456 – 474.
3. Baser M., Harmanci A., Kwak T. K. Generalized semicommutative rings and their extensions // Bull. Korean Math. –
2008. – 45. – P. 285 – 297.
4. Birkenmeier G. F., Kim J. K., Park J. K. Polynomial extensions of Baer and quasi-Baer rings // J. Pure and Appl.
Algebra. – 2001. – 159. – P. 25 – 42.
5. Clark E. W. Twisted matrix units semigroup algebras // Duke Math. J. – 1967. – 34. – P. 417 – 424.
6. Cui J., Chen J., On α-skew McCoy modules // Turk. J. Math. – 2012. – 36. – P. 217 – 229.
7. Hong C. Y., Kim N. K., Kwak T. K. On skew Armendariz rings // Communs Algebra. – 2003. – 31, № 1. – P. 103 – 122.
8. Huh C., Lee Y., Smoktunowicz A. Armendariz rings and semicommutative rings // Communs Algebra. – 2002. – 30. –
P. 751 – 761.
9. Nikmehr M. J. The structure of ideals over a monoid with applications // World Appl. Sci. J. – 2012. – 20, № 12. –
P. 1636 – 1641.
10. Nikmehr M. J., Fatahi F., Amraei H. Nil – Armendariz rings with applications to a monoid // World Appl. Sci. J. –
2011. – 13, № 12. – P. 2509 – 2514.
11. Tavallaee H. T., Nikmehr M. J., Pazoki M. Weak α-skew Armendariz ideals // Ukr. Math. J. – 2012. – 64, № 3. –
P. 456 – 469.
Received 02.09.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
|
| id | umjimathkievua-article-2212 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:49Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e0/1a9700c12c690718937db5765da576e0.pdf |
| spelling | umjimathkievua-article-22122019-12-05T10:26:31Z Generalized Semicommutative and Skew Armendariz Ideals Узагальнені напівкомутативні та косі iдеали Армендарiза Nikmehr, M. J. Нікмер, М. Дж. We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts. Узагальнено поняття напівкомутативних косих абелевих зведених та симетричних лівих iдеалiв Армендаріза та вивчено співвідношення між ними. Institute of Mathematics, NAS of Ukraine 2014-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2212 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 9 (2014); 1213–1222 Український математичний журнал; Том 66 № 9 (2014); 1213–1222 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2212/1425 https://umj.imath.kiev.ua/index.php/umj/article/view/2212/1426 Copyright (c) 2014 Nikmehr M. J. |
| spellingShingle | Nikmehr, M. J. Нікмер, М. Дж. Generalized Semicommutative and Skew Armendariz Ideals |
| title | Generalized Semicommutative and Skew Armendariz Ideals |
| title_alt | Узагальнені напівкомутативні та косі iдеали Армендарiза |
| title_full | Generalized Semicommutative and Skew Armendariz Ideals |
| title_fullStr | Generalized Semicommutative and Skew Armendariz Ideals |
| title_full_unstemmed | Generalized Semicommutative and Skew Armendariz Ideals |
| title_short | Generalized Semicommutative and Skew Armendariz Ideals |
| title_sort | generalized semicommutative and skew armendariz ideals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2212 |
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