Weak $\alpha$-skew Armendariz ideal
We introduce the concept of weak $\alpha$-skew Armendariz ideals and investigate their properties. Moreover, we prove that $I$ is a weak $\alpha$-skew Armendariz ideal if and only if $I[x]$ is a weak $\alpha$-skew Armendariz ideal. As a consequence, we show that $R$ is a weak $\alpha$-skew Armendari...
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2012
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| author | Nikmehr, M. J. Pazoki, M. Tavallaee, H. A. Нікмер, М. Дж. Пазокі, М. Таваллаее, Г. А. |
| author_facet | Nikmehr, M. J. Pazoki, M. Tavallaee, H. A. Нікмер, М. Дж. Пазокі, М. Таваллаее, Г. А. |
| author_sort | Nikmehr, M. J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:30:02Z |
| description | We introduce the concept of weak $\alpha$-skew Armendariz ideals and investigate their properties. Moreover, we prove that $I$ is a weak $\alpha$-skew Armendariz ideal if and only if $I[x]$ is a weak $\alpha$-skew Armendariz ideal. As a consequence, we show that $R$ is a weak $\alpha$-skew Armendariz ring if and only if $R[x]$ is a weak $\alpha$-skew Armendariz ring. |
| first_indexed | 2026-03-24T02:26:18Z |
| format | Article |
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UDC 512.5
H. A. Tavallaee (Karaj Branch, Islamic Azad Univ., Karaj, Iran),
M. J. Nikmehr (K. N. Toosi Univ. Technology, Tehran, Iran),
M. Pazoki (Karaj Branch, Islamic Azad Univ., Karaj, Iran)
WEAK α-SKEW ARMENDARIZ IDEAL
СЛАБКI α-КОСI IДЕАЛИ АРМЕНДАРIЗА
We introduce the concept of weak α-skew Armendariz ideals and investigate their properties. Moreover, we prove that I is
a weak α-skew Armendariz ideal if and only if I[x] is a weak α-skew Armendariz ideal. As a consequence, we show that
R is a weak α-skew Armendariz ring if and only if R[x] is a weak α-skew Armendariz ring.
Введено поняття слабких α-косих iдеалiв Армендарiза та дослiджено їх властивостi. Крiм того, доведено, що I є
слабким α-косим iдеалом Армендарiза тодi i тiльки тодi, коли I[x] є слабким α-косим iдеалом Армендарiза. Як
наслiдок, показано, що R є слабким α-косим кiльцем Армендарiза тодi i тiльки тодi, коли R[x] є слабким α-косим
кiльцем Армендарiза.
1. Introduction. In [11], Rege and Chhawchharia introduced the notion of an Armendariz ring.
They defined a ring R (associative with identity) to be an Armendariz ring if whenever polynomials
f(x) = a0 + a1x + . . . + amx
m, g(x) = b0 + b1x + . . . + bnx
n ∈ R[x] satisfy f(x)g(x) = 0, then
aibj = 0 for each i, j. (The converse is always true.) Some properties of Armendariz rings were
given in [1, 2, 5, 6, 11]. Throughout this paper R denotes an associative ring with identity. A ring
R is called semicommutative if for any a, b ∈ R, ab = 0 implies aRb = 0. The name Armendariz
ring was chosen because Armendariz [2] (Lemma 1) had noted that a reduced ring (i.e., a2 = 0
implies a = 0) satisfies this condition. Zhongkui Liu and Renyu Zhao [9] studied a generalization
of Armendariz ring, which is called weak Armendariz ring. A ring R is called weak Armendariz if
whenever f(x) = a0 + a1x + . . . + amx
m, g(x) = b0 + b1x + . . . + bnx
n ∈ R[x], with ai, bj ∈ R
satisfy f(x)g(x) = 0, then aibj is a nilpotent element of R for each i, j. They have shown that, if
R is a semicommutative ring, then the ring R[x] and the ring
R[x]
(xn)
, are weak Armendariz. For an
endomorphism α of a ring R, Hong, Kim, and Kwak [3] called R an α-skew Armendariz ring if
whenever polynomials f(x) = a0 + a1x + . . . + amx
m, g(x) = b0 + b1x + . . . + bnx
n ∈ R[x;α]
satisfy f(x)g(x) = 0, then aiαi(bj) = 0 for each i and j.
Recall from [10] that a one-sided ideal I of a ring R has the insertion of factors property (or
simply, IFP) if ab ∈ I implies aRb ⊆ I for a, b ∈ R. Observe that every completely semiprime ideal
(i.e., a2 ∈ I implies a ∈ I) of R has the IFP (or R is semicommutative).
For any positive integer n, we study in this paper the relationship between ideals of R which are
weak α-skew Armendariz and some ideals of the ring
Rn(R) =
a a12 . . . a1n
0 a . . . a2n
...
...
. . .
...
0 0 . . . a
∣∣∣∣∣∣∣∣∣∣∣∣
a, aij ∈ R, for all i, j
,
c© H. A. TAVALLAEE, M. J. NIKMEHR, M. PAZOKI, 2012
404 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
WEAK α-SKEW ARMENDARIZ IDEAL 405
the n-by-n upper triangular matrix ring over R and the ring
R[x]
(xn)
, where (xn) is the ideal generated
by xn. Also we show that, if I an ideal of R, then I is a weak α-skew Armendariz if and only if
I[x] is a weak α-skew Armendariz ideal.
2. On weak α-skew Armendariz ideals. For an ideal I of R put
√
I =
{
a ∈ R | an ∈ I for some non-negative integer n
}
.
Definition 2.1. Let α be an endomorphism of a ring R, an ideal I of R is said to be weak
α-skew Armendariz if whenever polynomials f(x) = a0 + a1x+ . . .+ amx
m, g(x) = b0 + b1x+ . . .
. . .+ bnx
n ∈ R[x] satisfy f(x)g(x) ∈ I[x] then aiαi(bj) ∈
√
I for all i, j.
Clearly, if I = 0 is a weak α-skew Armendariz ideal, then R is a weak α-skew Armendariz ring.
It is well-known that for a ring R and any positive integer n ≥ 2,
R[x]
(xn)
∼=
a0 a1 . . . an−1
0 a0 . . . an−2
...
...
. . .
...
0 0 . . . a0
∣∣∣∣∣∣∣∣∣∣∣∣
ai ∈ R, i = 0, 1, . . . , n− 1
,
where (xn) is the ideal of R[x] generated by xn.
We introduced a weak α-skew Armendariz ideal in the following example.
Example 2.1. Let R be a α-skew Armendariz ring and consider
S =
{(
a b
0 a
)∣∣∣∣∣ a, b ∈ R
}
.
It is clear that I =
{(
0 b
0 0
)∣∣∣∣∣ b ∈ R
}
is the ideal of S. Let f(x) = A0 + A1x + . . . + Anx
n,
g(x) = B0 + B1x + . . . + Bmx
m ∈ S[x], where Ai =
(
a0i a1i
0 a0i
)
, Bj =
(
b0j b1j
0 b0j
)
for
i = 0, . . . , n, j = 0, . . . ,m such that f(x)g(x) ∈ 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) ∈ I[x] thus α0(x)β0(x) = 0, also R is an α-skew Armendariz ring and hence
a0iα
i(b0j) = 0 for all i = 0, . . . , n, j = 0, . . . ,m. Thus Aiαi(Bj) ∈ I for all i = 0, . . . , n,
j = 0, . . . ,m. Therefore I is a weak α-skew Armendariz ideal.
Lemma 2.1. Let R be a ring and n ≥ 2 a positive integer. Let I0, I1, . . . , In−1 are ideals of
R, such that Ii ⊆ Ii+1, i = 0, 1, . . . , n− 2. Then
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
406 H. A. TAVALLAEE, M. J. NIKMEHR, M. PAZOKI
J =
a0 a1 . . . an−1
0 a0 . . . an−2
...
...
. . .
...
0 0 . . . a0
∣∣∣∣∣∣∣∣∣∣∣∣
ai ∈ Ii, i = 0, 1, . . . , n− 1
is an ideal of
R[x]
(xn)
.
Proof. It is straightforward.
We note that, in Proposition 2.1 and Theorem 2.1, I0 and J are ideals that mentioned in
Lemma 2.1.
Proposition 2.1. Let
Ai =
ai0 ai1 . . . ain−1
0 ai0 . . . ain−2
...
...
. . .
...
0 0 . . . ai0
, Bj =
bj0 bj1 . . . bjn−1
0 bj0 . . . bjn−2
...
...
. . .
...
0 0 . . . bj0
∈
R[x]
(xn)
such that (ai0α
i(bj0))
k ∈ I0 for any i, j and some integer k. Then (Aiα
i(Bj))
nk ∈ J.
Proof. We proceed by induction on n. Let n = 2. For a positive integer k, (Aiα
i(Bj))
k =
=
(
(ai0α
i(bj0))
k c
0 (ai0α
i(bj0))
k
)
and that
(Aiα
i(Bj))
2k =
(
(ai0α
i(bj0))
2k (ai0α
i(bj0))
kc+ c(ai0α
i(bj0))
k
0 (ai0α
i(bj0))
2k
)
.
Hence (Aiα
i(Bj))
2k ∈ J, since
(
ai0α
i(bj0)
)2k
, (ai0α
i(bj0))
kc+ c(ai0α
i(bj0))
k ∈ I0. Now, we have
Ai =
ai0 ai1 . . . ain−1
0 ai0 . . . ain−2
...
...
. . .
...
0 0 . . . ai0
∈
R[x]
(xn)
and
Bj =
bj0 bj1 . . . bjn−1
0 bj0 . . . bjn−2
...
...
. . .
...
0 0 . . . bj0
∈
R[x]
(xn)
,
such that (ai0α
i(bj0))
k ∈ I0 for some integer k. Consider
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
WEAK α-SKEW ARMENDARIZ IDEAL 407
(Aiα
i(Bj))
k =
(ai0α
i(bj0))
k c1 . . . cn−1
0 (ai0α
i(bj0))
k . . . cn−2
...
...
. . .
...
0 0 . . . (ai0α
i(bj0))
k
∈ J
and
(Aiα
i(Bj))
(n−1)k =
(ai0α
i(bj0))
(n−1)k d1 . . . dn−1
0 (ai0α
i(bj0))
(n−1)k . . . dn−2
...
...
. . .
...
0 0 . . . (ai0α
i(bj0))
(n−1)k
∈ J.
By the induction hypothesis all d,is, except dn−1, are in I0. Let x = (ai0α
i(bj0))
kdn−1 + c1dn−2 + . . .
. . .+ cn−1(a
i
0α
i(bj0))
(n−1)k. Hence
(Aiα
i(Bj))
nk =
(ai0α
i(bj0))
nk y1 . . . x
0 (ai0α
i(bj0))
nk . . . yn−2
...
...
. . .
...
0 0 . . . (ai0α
i(bj0))
nk
∈ J,
since (ai0α
i(bj0))
nk, x all y,is are in I0.
Proposition 2.1 is proved.
Theorem 2.1. I0 is a weak α-skew Armendariz ideal if and only if J is a weak α-skew
Armendariz ideal.
Proof. (⇒) Let f(y) = A0 + A1y + . . . + Amy
m, g(y) = B0 + B1y + . . . + Bty
t ∈ R[x]
(xn)
[y],
such that f(y)g(y) ∈ J [y]. Let
Ai =
ai0 ai1 . . . ain−1
0 ai0 . . . ain−2
...
...
. . .
...
0 0 . . . ai0
, Bj =
bj0 bj1 . . . bjn−1
0 bj0 . . . bjn−2
...
...
. . .
...
0 0 . . . bj0
for i = 0, 1, . . . ,m, j = 0, 1, . . . , t. Let f0 = a00 +a10y+ . . .+am0 y
m and g0 = b00 + b10y+ . . .+ bt0y
t.
Then f0g0 ∈ I0[y]. Since I0 is weak α-skew Armendariz, there exists k > 0, such that (ai0α
ibj0)
k ∈ I0
for each i, j. Then (Aiα
i(Bj))
nk ∈ J for all i, j, by Proposition 2.1. Therefore J is weak α-skew
Armendariz.
(⇐) Clear.
Theorem 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
408 H. A. TAVALLAEE, M. J. NIKMEHR, M. PAZOKI
It can be simply proved if I be an ideal of ring R, then Tn(I) will also be an ideal of ring Tn(R),
where Tn(I) is an upper triangle matrix. By the following example we show that T2(pZ) is a weak
α-skew Armendariz ideal.
Example 2.2. Let pZ be a prime ideal of Z and α : pZ → pZ be an endomorphism. Then
T2(pZ) is a weak α-skew Armendariz ideal.
Let
γ(x) =
n∑
i=0
(
γi0 γi1
0 γi2
)
xi, β(x) =
n∑
j=0
(
βj0 βj1
0 βj2
)
xj ∈ T2(Z)[x],
such that γ(x)β(x) ∈ T2(pZ)[x]. Let
γ(x) =
(
γ0(x) γ1(x)
0 γ2(x)
)
, β(x) =
(
β0(x) β1(x)
0 β2(x)
)
.
Thus (
γ0(x) γ1(x)
0 γ2(x)
)(
β0(x) β1(x)
0 β2(x)
)
∈ T2(pZ)[x],
and hence we have
γ0(x)β0(x) ∈ pZ[x],
γ0(x)β1(x) + γ1(x)β2(x) ∈ pZ[x],
γ2(x)β2(x) ∈ pZ[x].
Since pZ[x] is a prime ideal of Z, two cases happen for polynomials,
Case 1. γ0(x), γ1(x), γ2(x) ∈ pZ[x], therefore(
γi0 γi1
0 γi2
)
αi
(
βj0 βj1
0 βj2
)
∈ T2(pZ).
Case 2. γ0(x), β2(x) ∈ pZ[x], therefore(
γi0 γi1
0 γi2
)
αi
(
βj0 βj1
0 βj2
)
∈ T2(pZ).
Thus T2(pZ) is a weak α-skew Armendariz ideal.
Let α be an endomorphism of a ring R, Mn(R) be the n × n full matrix ring over R and
α : Mn(R) −→Mn(R) defined by α((aij)) = (α(aij)). Then α is an endomorphism of Mn(R).
Theorem 2.2. I0 is a weak α-skew Armendariz ideal if and only if J is a weak α-skew
Armendariz ideal.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
WEAK α-SKEW ARMENDARIZ IDEAL 409
Proof. (⇒) Let f(y) = A0 + A1y + . . .+ Apy
p, g(y) = B0 +B1y + . . .+Bqy
q ∈ R[x]
(xn)
[y;α]
satisfying f(y)g(y) ∈ J [y], where
Ai =
ai ai12 ai1n . . . ai1n
0 ai ai23 . . . ai2n
0 0 ai . . . ai3n
...
...
...
. . .
...
0 0 0 . . . ai
and Bj =
bi bi12 bi13 . . . bi1n
0 bi bi23 . . . bi2n
0 0 bi . . . bi3n
...
...
...
. . .
...
0 0 0 . . . bi
for i = 0, 1, . . . , p, j = 0, 1, . . . , q. Let f0 = a00+a10y+. . .+ap0y
p and g0 = b00+b10y+. . .+bq0y
q. Then
f0g0 ∈ I0[y]. Since I0 is weak α-skew Armendariz, there exists k > 0, such that (ai0α
i(bj0))
k ∈ I0
for each i, j. Then (Aiα
i(Bj))
nk ∈ J for all i, j, by Proposition 2.1 and α((aij)) = (α(aij)).
Therefore J is a weak α-skew Armendariz ideal.
(⇐) Clear.
Theorem 2.2 is proved.
For the case of weak α-skew Armendariz ideal, we have the following result.
Theorem 2.3. Let α be an endomorphism of a ring R and αt = 1R for some positive integer
t. Then I is a weak α-skew Armendariz ideal if and only if I[x] is a weak α-skew Armendariz ideal.
Proof. (⇒) Assume that I is a weak α-skew Armendariz ideal. Suppose that p(y) = f0(x) +
+f1(x)y+. . .+fm(x)ym and q(y) = g0(x)+g1(x)y+. . .+gn(x)yn are in R[x][y;α] with p(y)q(y) ∈
∈ I[x][y;α]. We also let fi(x) = ai0 + ai1x+ . . .+ aiωix
ωi and gj(x) = bj0 + bj1x+ . . .+ bjυjx
υj
for any 0 ≤ i ≤ m, 0 ≤ j ≤ n, where ai0, ai1, . . . , aiωi , bj0, bj1, . . . , bjυj ∈ R. We claim that
fi(x)αi(gj(x)) ∈
√
I[x] for all 0 ≤ i ≤ m and 0 ≤ j ≤ n. Take a positive integer k such that
k > deg(f0(x)) + deg(f1(x)) + . . .+ deg(fm(x)) + deg(g0(x)) + deg(g1(x)) + . . .+ deg(gn(x)),
where the degree is as a polynomials in R[x] and the degree of the zero polynomial is taken to be 0.
Since p(y)q(y) ∈ I[x][y;α], we have
f0(x)g0(x) ∈ I[x],
f0(x)g1(x) + f1(x)α(g0(x)) ∈ I[x],
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
fm(x)αm(gn(x)) ∈ I[x].
(1)
Now put
f(x) = f0(x
t) + f1(x
t)xtk+1 + f2(x
t)x2tk+2 + . . .+ fm(xt)xmtk+m,
g(x) = g0(x
t) + g1(x
t)xtk+1 + g2(x
t)x2tk+2 + . . .+ gn(xt)xntk+n.
(2)
Note that αt = 1R, then f(x)g(x) = f0(x
t)g0(x
t) + (f0(x
t)g1(x
t) + f1(x
t)α(g0(x
t)))xtk+1 + . . .
. . .+ fm(xt)αm(gn(xt))x(m+n)(tk+1). Using (1) and αt = 1R, we have f(x)g(x) ∈ I[x;α]. On the
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
410 H. A. TAVALLAEE, M. J. NIKMEHR, M. PAZOKI
other hand, from (2) we have
f(x)g(x) =
(
a00 + a01x
t + . . .+ a0ω0x
ω0t + a10x
tk+1 + a11x
tk+t+1 + . . .
. . .+ a1ω1x
tk+ω1t+1 + . . .+ am0x
mtk+m + am1x
mtk+t+m + . . .+ amωmx
mtk+ωmt+m
)
×
×
(
b00 + b01x
t + . . .+ b0υ0x
υ0t + b10x
tk+1 + b11x
tk+t+1 + . . .+ b1υ1x
tk+υ1t+1 + . . .
. . .+ bn0x
ntk+n + bn1x
ntk+t+n + . . .+ bnυnx
ntk+υnt+n
)
∈ I[x;α].
Since I is a weak α-skew Armendariz ideal and αt = 1R, so aiuαi(bjv) = aiuα
itk+ut+i(bjv) ∈
√
I
for all 0 ≤ i ≤ m and 0 ≤ j ≤ n, u ∈ {0, 1, . . . , ω0, . . . , ωm}, v ∈ {0, 1, . . . , υ0, . . . , υn}. So
we have fi(xt)αi(gj(xt)) ∈
√
I[x] for all 0 ≤ i ≤ m and 0 ≤ j ≤ n. Now it is easy to see that
fi(x)αi(gj(x)) ∈
√
I[x] for all 0 ≤ i ≤ m and 0 ≤ j ≤ n. Hence I[x] is weak α-skew Armendariz.
(⇐) Obviously, if I[x] is weak α-skew Armendariz, then I is weak α-skew Armendariz.
Theorem 2.3 is proved.
Using Theorem 2.3, we have the following result.
Corollary 2.1. Let R be a ring. Then R is weak α-skew Armendariz if and only if R[x] is weak
α-skew Armendariz.
Before stating Proposition 2.3, we need the following.
Proposition 2.2. Suppose that there exists a classical right quotient ring Q of a ring R con-
sisting of central elements. If I is IFP, then QI is IFP.
Proof. Let αβ ∈ QI with α = u−1a, β = υ−1b in Q such that, u, υ ∈ R and a, b ∈ R. Since
Q is contained in the center of R, we have (uυ)−1ab = (u−1υ−1)ab = u−1aυ−1b = αβ ∈ QI, so
ab ∈ I, and hence arb ∈ I for all r ∈ R because I is IFP. Now for γ = ω−1r with ω ∈ R and
r ∈ R, αγβ = (uωυ)−1arb ∈ QI. Therefore QI is IFP.
Proposition 2.2 is proved.
A ring 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.
Let α be an automorphism of a ring R. Suppose that there exists the classical left quotient Q
of 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 2.3. Suppose that there exists the classical left quotient Q of a ring R. If I is IFP,
then I is weak α-skew Armendariz if and only if QI is weak α-skew Armendariz.
Proof. Suppose that I is weak α-skew Armendariz. Let f(x) = s−10 a0+s−11 a1x+. . .+s−1m amx
m
and g(x) = t−10 b0 + t−11 b1x + . . . + t−1n bnx
n ∈ QI[x;α] such that f(x)g(x) ∈ QI[x]. Let C
be a left denominator set. There exist s, t ∈ C and a′i, b
′
j ∈ R such that s−1i ai = s−1a′i and
t−1j 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) ∈ QI[x]. It follows that (a′0 + a′1x+ . . .+ a′mx
m)t−1(b′0 + b′1x+ . . .+ b′nx
n) ∈
∈ QI[x]. Thus (a′0t
−1 +a′1(α(t))−1x+ . . .+a′m(αm(t))−1xm)(b′0 + b′1x+ . . .+ b′nx
n) ∈ 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
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
WEAK α-SKEW ARMENDARIZ IDEAL 411
t′−1(a′′0 + a′′1x + . . . + a′′mx
m)(b′0 + b′1x + . . . + b′nx
n) ∈ QI[x]. We have that (a′′0 + a′′1x + . . .
. . .+ a′′mx
m)(b′0 + b′1x+ . . .+ b′nx
n) ∈ I[x]. Since I is weak α-skew Armendariz, so a′′i α
i(b′j) ∈
√
I
for all i and j. Suppose that (a′′i α
i(b′j))
nij ∈ I. Since I is IFP, QI is IFP. Then (t′−1(a′′i α
i(b′j)))
nij ∈
∈ QI . So (a′iα
i(t−1b′j))
nij = (a′i(α
i(t))−1αi(b′j))
nij = ((t′−1a′′i )α
i(b′j))
nij ∈ QI. Similarly we
have (s−1i a′i)(α
i(t−1j b′j))
nij = (s−1a′i)(α
i(t−1b′j))
nij ∈ QI. Therefore QI is weak α-skew Armen-
dariz. The converse is clear.
Proposition 2.3 is proved.
We study the relationship between ideals of R which are weak α-skew Armendariz with some
ideals of the ring Rn(R).
Lemma 2.2. Let I, Iij be ideals of R such that I ⊆ Iij ⊆ Iis for 1 ≤ i < j ≤ s ≤ n, and
Ipq ⊆ Ilq for q = 3, . . . , n, 2 ≤ l ≤ p ≤ n. Then
J =
a a12 . . . a1n
0 a . . . a2n
...
...
. . .
...
0 0 . . . a
∣∣∣∣∣∣∣∣∣ a ∈ I, aij ∈ Iij
is an ideal of Rn(R).
Proof. It is straightforward.
In Proposition 2.4 and Theorem 2.4, I and J are ideals that mentioned in Lemma 2.2.
Proposition 2.4. Let
Ai =
ai ai1 . . . ain−1
0 ai . . . ain−2
...
...
. . .
...
0 0 . . . ai
, Bj =
bj bj1 . . . bjn−1
0 bj . . . bjn−2
...
...
. . .
...
0 0 . . . bj
∈ Rn(R)
such that (ai0α
i(bj0))
k ∈ I for any i, j and some integer k. Then (Aiα
i(Bj))
nk ∈ J.
Proof. We proceed by induction on n. Let n = 2. For a positive integer k,
(Aiα
i(Bj))
k =
(
(aiαi(bj))k c
0 (aiαi(bj))k
)
and that
(Aiα
i(Bj))
2k =
(
(aiαi(bj))2k (aiαi(bj))kc+ c(aiαi(bj))k
0 (aiαi(bj))2k
)
.
Hence (Aiα
i(Bj)) ∈ J, since
(
aiαi(bj)
)2k
, (aiαi(bj))kc+ c(aiαi(bj))k ∈ I. Now, let
Ai =
ai ai1 . . . ain−1
0 ai . . . ain−2
...
...
. . .
...
0 0 . . . ai
∈ Rn(R)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
412 H. A. TAVALLAEE, M. J. NIKMEHR, M. PAZOKI
and
Bj =
bj bj1 . . . bjn−1
0 bj . . . bjn−2
...
...
. . .
...
0 0 . . . bj
∈ Rn(R)
such that (aiαi(bj))k ∈ I for some integer k. Consider
(Aiα
i(Bj))
k =
(aiαi(bj))k c1 . . . cn−1
0 (aiαi(bj))k . . . cn−2
...
...
. . .
...
0 0 . . . (aiαi(bj))k
∈ J
and
(Aiα
i(Bj))
(n−1)k =
(aiαi(bj))(n−1)k b1 . . . dn−1
0 (aiαi(bj))(n−1)k . . . dn−2
...
...
. . .
...
0 0 . . . (aiαi(bj))(n−1)k
∈ J,
by the induction hypothesis all d,is, except dn−1, are in I. Let x = (aiαi(bj))kdn−1 + c1dn−2 + . . .
. . .+ cn−1(a
iαi(bj))(n−1)k. Hence
(Aiα
i(Bj))
nk =
(aiαi(bj))nk y1 . . . x
0 (aiαi(bj))nk . . . yn−2
...
...
. . .
...
0 0 . . . (aiαi(bj))nk
∈ J,
since (aiαi(bj))nk, x all y,is are in I.
Proposition 2.4 is proved.
Theorem 2.4. I is a weak α-skew Armendariz ideal if and only if J is a weak α-skew Armen-
dariz ideal.
Proof. (⇒) Let f(y) = A0 + A1y + . . . + Amy
m, g(y) = B0 + B1y + . . . + Bty
t ∈ Rn(R),
such that f(y)g(y) ∈ J [y]. Let
Ai =
ai0 ai1 . . . ain−1
0 ai0 . . . ain−2
...
...
. . .
...
0 0 . . . ai0
, Bj =
bj0 bj1 . . . bjn−1
0 bj0 . . . bjn−2
...
...
. . .
...
0 0 . . . bj0
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
WEAK α-SKEW ARMENDARIZ IDEAL 413
for i = 0, 1, . . . ,m, j = 0, 1, . . . , t. Let f0 = a00 +a10y+ . . .+am0 y
m and g0 = b00 + b10y+ . . .+ bt0y
t.
Then f0g0 ∈ I[y]. Since I is weak α-skew Armendariz, there exists k > 0, such that (ai0α
ibj0)
k ∈ I
for each i, j. Then (Aiα
i(Bj))
nk ∈ J for all i, j, by Proposition 2.4. Therefore J is weak α-skew
Armendariz.
(⇐) Clear.
Theorem 2.4 is proved.
Corollary 2.2. A ring R is weak α-skew Armendariz if and only if for any positive integer n,
Rn(R) is weak α-skew Armendariz.
Proof. It follows from Theorem 2.4.
Now, we prove the Theorem 2.4 for α : Mn(R) −→Mn(R).
Theorem 2.5. I is a weak α-skew Armendariz ideal if and only if J is a weak α-skew Armen-
dariz ideal.
Proof. (⇒) Let f(y) = A0 + A1y + . . . + Apy
p, g(y) = B0 + B1y + . . . + Bqy
q ∈ Rn(R)
satisfying f(y)g(y) ∈ J [y], where
Ai =
ai ai12 ai1n . . . ai1n
0 ai ai23 . . . ai2n
0 0 ai . . . ai3n
...
...
...
. . .
...
0 0 0 . . . ai
and Bj =
bi bi12 bi13 . . . bi1n
0 bi bi23 . . . bi2n
0 0 bi . . . bi3n
...
...
...
. . .
...
0 0 0 . . . bi
for i = 0, 1, . . . , p, j = 0, 1, . . . , q. Let f0 = a00+a10y+. . .+ap0y
p and g0 = b00+b10y+. . .+bq0y
q. Then
f0g0 ∈ I[y]. Since I is weak α- skew Armendariz, there exists k > 0, such that (aiαi(bj))k ∈ I for
each i, j. Then (Aiα
i(Bj))
nk ∈ J for all i, j, by Proposition 2.4, and α(aij) = (α(aij)). Therefore
J is a weak α-skew Armendariz ideal.
(⇐) Clear.
Theorem 2.5 is proved.
Theorem 2.6. Let R be a ring and I, J be ideals of R. If I ⊆
√
J and
I + J
I
is weak α-skew
Armendariz, then J is a weak α-skew Armendariz ideal.
Proof. Let f(x) = Σm
i=0aix
i, g(x) = Σt
j=0bjx
j ∈ R[x] such that f(x)g(x) ∈ J [x]. Then
(
Σm
i=0aix
i
)(
Σt
j=0bjx
j
)
∈ I + J
I
[x].
Thus
(
aiα
ibj
)nij ∈ I + J
I
for some positive integer nij . Hence (aiα
ibj)
nij ∈ I + J, and so(
aiα
ibj
)nij ∈ J, since I ⊆
√
J. Therefore J is weak α-skew Armendariz.
Theorem 2.6 is proved.
The following is an immediate corollary of Theorem 2.6.
Corollary 2.3. Let R be a ring and I an ideal of R such that
R
I
is weak α-skew Armendariz. If
I ⊆ nil(R), then R is weak α-skew Armendariz.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
414 H. A. TAVALLAEE, M. J. NIKMEHR, M. PAZOKI
Lemma 2.3. Let Irt be ideals of R such that Irt⊆ Irs for 1 ≤ r ≤ t ≤ s ≤ n, and Ipq ⊆ Ilq
for q = 2, . . . , n, 1 ≤ l ≤ p ≤ n. Then
J =
a11 a12 . . . a1n
0 a22 . . . a2n
...
...
. . .
...
0 0 . . . ann
∣∣∣∣∣∣∣∣∣∣∣∣
art ∈ Irt, 1 ≤ r, t ≤ n
is an ideal of Tn(R).
Proof. It is straightforward.
In Corollaries 2.4 and 2.5 I ,rts are ideals that mentioned in Lemma 2.3. By a similar argument as
used in the proof of Proposition 2.1 and Theorem 2.1, one can prove Corollaries 2.4 and 2.5.
Corollary 2.4. Let
Ai =
ai11 ai12 . . . ai1n
0 ai22 . . . ai2n
...
...
. . .
...
0 0 . . . ainn
, Bj =
bj11 bj12 . . . bj1n
0 bj22 . . . bj2n
...
...
. . .
...
0 0 . . . bjnn
∈ Tn(R)
such that (airrα
i(bjrr))k ∈ Irr for some positive integer k and r = 1, . . . , n. Then(
(Aiα
i(Bj))
2k+1
)n−1 ∈ J.
Corollary 2.5. J is a weak α-skew Armendariz ideal if and only if all Irr are weak α-skew
Armendariz ideal for r = 1, . . . , n.
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Received 18.10.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
|
| id | umjimathkievua-article-2585 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:18Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0b/cac0a087c4cbf10a82e4b5943a53370b.pdf |
| spelling | umjimathkievua-article-25852020-03-18T19:30:02Z Weak $\alpha$-skew Armendariz ideal Слабкi $\alpha$-косi iдеали Армендарiза Nikmehr, M. J. Pazoki, M. Tavallaee, H. A. Нікмер, М. Дж. Пазокі, М. Таваллаее, Г. А. We introduce the concept of weak $\alpha$-skew Armendariz ideals and investigate their properties. Moreover, we prove that $I$ is a weak $\alpha$-skew Armendariz ideal if and only if $I[x]$ is a weak $\alpha$-skew Armendariz ideal. As a consequence, we show that $R$ is a weak $\alpha$-skew Armendariz ring if and only if $R[x]$ is a weak $\alpha$-skew Armendariz ring. Введено поняття слабких $\alpha$-косих iдеалiв Армендарiза та дослiджено їх властивостi. Крiм того, доведено, що $I$ є слабким $\alpha$-косим iдеалом Армендарiза тодi i тiльки тодi, коли $I[x]$ є слабким $\alpha$-косим iдеалом Армендарiза. Як наслiдок, показано, що $R$ є слабким $\alpha$-косим кiльцем Армендарiза тодi i тiльки тодi, коли $R[x]$ є слабким $\alpha$-косим кiльцем Армендарiза. Institute of Mathematics, NAS of Ukraine 2012-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2585 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 3 (2012); 404-414 Український математичний журнал; Том 64 № 3 (2012); 404-414 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2585/1929 https://umj.imath.kiev.ua/index.php/umj/article/view/2585/1930 Copyright (c) 2012 Nikmehr M. J.; Pazoki M.; Tavallaee H. A. |
| spellingShingle | Nikmehr, M. J. Pazoki, M. Tavallaee, H. A. Нікмер, М. Дж. Пазокі, М. Таваллаее, Г. А. Weak $\alpha$-skew Armendariz ideal |
| title | Weak $\alpha$-skew Armendariz ideal |
| title_alt | Слабкi $\alpha$-косi iдеали Армендарiза |
| title_full | Weak $\alpha$-skew Armendariz ideal |
| title_fullStr | Weak $\alpha$-skew Armendariz ideal |
| title_full_unstemmed | Weak $\alpha$-skew Armendariz ideal |
| title_short | Weak $\alpha$-skew Armendariz ideal |
| title_sort | weak $\alpha$-skew armendariz ideal |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2585 |
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