Decompositions of Quotient Rings and m-Commuting Maps
Transcript of Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps
Decompositions of Quotient Rings and
m-Commuting Maps
Chih-Whi Chen
Department of MathematicsNational Taiwan University
July 14th, 2011
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
R : semiprime ring.
Q : symmetric Martindale quotient ring of R .
C : extended centroid of R .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
R : semiprime ring.
Q : symmetric Martindale quotient ring of R .
C : extended centroid of R .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
![Page 4: Decompositions of Quotient Rings and m-Commuting Maps](https://reader031.fdocuments.us/reader031/viewer/2022021007/62039483da24ad121e4b0ffe/html5/thumbnails/4.jpg)
Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
R : semiprime ring.
Q : symmetric Martindale quotient ring of R .
C : extended centroid of R .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
![Page 5: Decompositions of Quotient Rings and m-Commuting Maps](https://reader031.fdocuments.us/reader031/viewer/2022021007/62039483da24ad121e4b0ffe/html5/thumbnails/5.jpg)
Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
Z{X̂} the free Z-algebra in noncommutativeindeterminates in X̂ := {X ,X1,X2, . . .}.
R satisfies f (X1, . . . ,Xt) ∈ Z{X̂} if
f (r1, . . . , rt) = 0 for all ri ∈ R .
R is called faithful f -free if I doesn’t satisfy f for everynonzero I C R .
Sk(X1, . . . ,Xk) :=∑
σ∈Sym(k)
(−1)σXσ(1) · · ·Xσ(k).
For example, S2(X1,X2) = X1X2 − X2X1.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
Z{X̂} the free Z-algebra in noncommutativeindeterminates in X̂ := {X ,X1,X2, . . .}.R satisfies f (X1, . . . ,Xt) ∈ Z{X̂} if
f (r1, . . . , rt) = 0 for all ri ∈ R .
R is called faithful f -free if I doesn’t satisfy f for everynonzero I C R .
Sk(X1, . . . ,Xk) :=∑
σ∈Sym(k)
(−1)σXσ(1) · · ·Xσ(k).
For example, S2(X1,X2) = X1X2 − X2X1.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
Z{X̂} the free Z-algebra in noncommutativeindeterminates in X̂ := {X ,X1,X2, . . .}.R satisfies f (X1, . . . ,Xt) ∈ Z{X̂} if
f (r1, . . . , rt) = 0 for all ri ∈ R .
R is called faithful f -free if I doesn’t satisfy f for everynonzero I C R .
Sk(X1, . . . ,Xk) :=∑
σ∈Sym(k)
(−1)σXσ(1) · · ·Xσ(k).
For example, S2(X1,X2) = X1X2 − X2X1.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
![Page 8: Decompositions of Quotient Rings and m-Commuting Maps](https://reader031.fdocuments.us/reader031/viewer/2022021007/62039483da24ad121e4b0ffe/html5/thumbnails/8.jpg)
Notations Decomposition of Q m-commuting maps Semiprimeness Polynomial identities
Z{X̂} the free Z-algebra in noncommutativeindeterminates in X̂ := {X ,X1,X2, . . .}.R satisfies f (X1, . . . ,Xt) ∈ Z{X̂} if
f (r1, . . . , rt) = 0 for all ri ∈ R .
R is called faithful f -free if I doesn’t satisfy f for everynonzero I C R .
Sk(X1, . . . ,Xk) :=∑
σ∈Sym(k)
(−1)σXσ(1) · · ·Xσ(k).
For example, S2(X1,X2) = X1X2 − X2X1.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 1 (T. Kosan, T.-K. Lee, Y. Zhou)
Let f (X ) ∈ Z{X̂}. Then there is a decomposition
Q = Q1 ⊕ Q2
such that the following hold:
1 Q1 is a ring satisfying f .
2 Q2 is a faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 1 (T. Kosan, T.-K. Lee, Y. Zhou)
Let f (X ) ∈ Z{X̂}. Then there is a decomposition
Q = Q1 ⊕ Q2
such that the following hold:
1 Q1 is a ring satisfying f .
2 Q2 is a faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 1 (T. Kosan, T.-K. Lee, Y. Zhou)
Let f (X ) ∈ Z{X̂}. Then there is a decomposition
Q = Q1 ⊕ Q2
such that the following hold:
1 Q1 is a ring satisfying f .
2 Q2 is a faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 2 (C. -W. Chen and T. -K. Lee, 2011)
Let n ≥ 2 and f (X ) = X nh(X ) where h(X ) is a polynomialover Z with h(0) = ±1. Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a ring satisfying S2n−2.
2 Q2 satisfies f and Q2∼= Mn(E ) where E is a commutative
self-injective ring such that, for some fixed integer q > 1,xq = x for all x ∈ E .
3 Q3 is a both faithful S2n−2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 2 (C. -W. Chen and T. -K. Lee, 2011)
Let n ≥ 2 and f (X ) = X nh(X ) where h(X ) is a polynomialover Z with h(0) = ±1. Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a ring satisfying S2n−2.
2 Q2 satisfies f and Q2∼= Mn(E ) where E is a commutative
self-injective ring such that, for some fixed integer q > 1,xq = x for all x ∈ E .
3 Q3 is a both faithful S2n−2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 2 (C. -W. Chen and T. -K. Lee, 2011)
Let n ≥ 2 and f (X ) = X nh(X ) where h(X ) is a polynomialover Z with h(0) = ±1. Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a ring satisfying S2n−2.
2 Q2 satisfies f and Q2∼= Mn(E ) where E is a commutative
self-injective ring such that, for some fixed integer q > 1,xq = x for all x ∈ E .
3 Q3 is a both faithful S2n−2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Motivations Main results
Theorem 2 (C. -W. Chen and T. -K. Lee, 2011)
Let n ≥ 2 and f (X ) = X nh(X ) where h(X ) is a polynomialover Z with h(0) = ±1. Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a ring satisfying S2n−2.
2 Q2 satisfies f and Q2∼= Mn(E ) where E is a commutative
self-injective ring such that, for some fixed integer q > 1,xq = x for all x ∈ E .
3 Q3 is a both faithful S2n−2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Definition 3
An additive map ψ : R → R is said to be m-commuting if
[ψ(x), xm] = 0 for all x ∈ R .
Definition 4
D(R) := {δ ∈ Der(Q) | I δ ⊆ I , for some I Cess. R}.A derivaition word: ∆ = δ1δ2 · · · δt , each δk ∈ D(R).
A map by linear generalized differential polynomial:
X →∑i
∑j
aijX∆j bij
where aij , bij ∈ Q and derivation words ∆j .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Definition 3
An additive map ψ : R → R is said to be m-commuting if
[ψ(x), xm] = 0 for all x ∈ R .
Definition 4
D(R) := {δ ∈ Der(Q) | I δ ⊆ I , for some I Cess. R}.
A derivaition word: ∆ = δ1δ2 · · · δt , each δk ∈ D(R).
A map by linear generalized differential polynomial:
X →∑i
∑j
aijX∆j bij
where aij , bij ∈ Q and derivation words ∆j .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Definition 3
An additive map ψ : R → R is said to be m-commuting if
[ψ(x), xm] = 0 for all x ∈ R .
Definition 4
D(R) := {δ ∈ Der(Q) | I δ ⊆ I , for some I Cess. R}.A derivaition word: ∆ = δ1δ2 · · · δt , each δk ∈ D(R).
A map by linear generalized differential polynomial:
X →∑i
∑j
aijX∆j bij
where aij , bij ∈ Q and derivation words ∆j .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Definition 3
An additive map ψ : R → R is said to be m-commuting if
[ψ(x), xm] = 0 for all x ∈ R .
Definition 4
D(R) := {δ ∈ Der(Q) | I δ ⊆ I , for some I Cess. R}.A derivaition word: ∆ = δ1δ2 · · · δt , each δk ∈ D(R).
A map by linear generalized differential polynomial:
X →∑i
∑j
aijX∆j bij
where aij , bij ∈ Q and derivation words ∆j .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
![Page 20: Decompositions of Quotient Rings and m-Commuting Maps](https://reader031.fdocuments.us/reader031/viewer/2022021007/62039483da24ad121e4b0ffe/html5/thumbnails/20.jpg)
Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 5 (T.-K. Lee, K.-S. Liu, W.-K. Shiue, 2004)
Let R be a noncommutative prime ring. R � M2(GF(2)).ψ : Q → Q satisfying [ψ(x), xm] = 0 for all x ∈ R, where ψdefined by a linear generalized differential polynomial. Then
ψ(x) = λx + µ(x)
for all x ∈ R, where λ ∈ C and µ : R → C .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Lemma 6 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then the following are equivalent:
1 R is both faithful f -free and faithful S2-free.
2 For any minimal prime ideal P of Q, neither Q/P iscommutative, nor Q/P ∼= M2(GF(2)).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Lemma 6 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then the following are equivalent:
1 R is both faithful f -free and faithful S2-free.
2 For any minimal prime ideal P of Q, neither Q/P iscommutative, nor Q/P ∼= M2(GF(2)).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Lemma 6 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then the following are equivalent:
1 R is both faithful f -free and faithful S2-free.
2 For any minimal prime ideal P of Q, neither Q/P iscommutative, nor Q/P ∼= M2(GF(2)).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 7 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such tha the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 Q3 is a both faithful S2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 7 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such tha the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 Q3 is a both faithful S2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 7 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such tha the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 Q3 is a both faithful S2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 7 (C.-W. Chen and T.-K. Lee, 2011)
Letf (X ) = X 2(X 2 + 1)(X 2 + X + 1).
Then there is a decomposition
Q = Q1 ⊕ Q2 ⊕ Q3
such tha the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 Q3 is a both faithful S2-free and faithful f -free ring.
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 8 (C.-W. Chen and T.-K. Lee, 2011)
Suppose that ψ : Q → Q satisfying [ψ(x), xm] = 0 for allx ∈ R, where ψ defined by a linear generalized differentialpolynomial. Then
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 ψ(x) = λx + µ(x) for all x ∈ Q3, where λ ∈ C andµ : Q3 → C .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 8 (C.-W. Chen and T.-K. Lee, 2011)
Suppose that ψ : Q → Q satisfying [ψ(x), xm] = 0 for allx ∈ R, where ψ defined by a linear generalized differentialpolynomial. Then
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 ψ(x) = λx + µ(x) for all x ∈ Q3, where λ ∈ C andµ : Q3 → C .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 8 (C.-W. Chen and T.-K. Lee, 2011)
Suppose that ψ : Q → Q satisfying [ψ(x), xm] = 0 for allx ∈ R, where ψ defined by a linear generalized differentialpolynomial. Then
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 ψ(x) = λx + µ(x) for all x ∈ Q3, where λ ∈ C andµ : Q3 → C .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
Theorem 8 (C.-W. Chen and T.-K. Lee, 2011)
Suppose that ψ : Q → Q satisfying [ψ(x), xm] = 0 for allx ∈ R, where ψ defined by a linear generalized differentialpolynomial. Then
Q = Q1 ⊕ Q2 ⊕ Q3
such that the following hold:
1 Q1 is a commutative ring.
2 Q2∼= M2(E ) where E is a complete Boolean algebra.
3 ψ(x) = λx + µ(x) for all x ∈ Q3, where λ ∈ C andµ : Q3 → C .
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
In Case Q2 := M2(GF(2)).
Example 9 (T.-K. Lee, K.-S. Liu, W.-K. Shiue ,2004)
R := M2(GF(2)). f : R → R defined by
f
((α βγ δ
))=
(α + γ 0
0 β + δ
)
for
(α βγ δ
)∈ R . Then f is a GF(2)-linear map.
1 [f (x), x6] = 0 for all x ∈ R .
2
[f
((0 10 0
)),
(0 10 0
)]=
(0 10 0
).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
In Case Q2 := M2(GF(2)).
Example 9 (T.-K. Lee, K.-S. Liu, W.-K. Shiue ,2004)
R := M2(GF(2)). f : R → R defined by
f
((α βγ δ
))=
(α + γ 0
0 β + δ
)
for
(α βγ δ
)∈ R .
Then f is a GF(2)-linear map.
1 [f (x), x6] = 0 for all x ∈ R .
2
[f
((0 10 0
)),
(0 10 0
)]=
(0 10 0
).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
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Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
In Case Q2 := M2(GF(2)).
Example 9 (T.-K. Lee, K.-S. Liu, W.-K. Shiue ,2004)
R := M2(GF(2)). f : R → R defined by
f
((α βγ δ
))=
(α + γ 0
0 β + δ
)
for
(α βγ δ
)∈ R . Then f is a GF(2)-linear map.
1 [f (x), x6] = 0 for all x ∈ R .
2
[f
((0 10 0
)),
(0 10 0
)]=
(0 10 0
).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
![Page 35: Decompositions of Quotient Rings and m-Commuting Maps](https://reader031.fdocuments.us/reader031/viewer/2022021007/62039483da24ad121e4b0ffe/html5/thumbnails/35.jpg)
Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
In Case Q2 := M2(GF(2)).
Example 9 (T.-K. Lee, K.-S. Liu, W.-K. Shiue ,2004)
R := M2(GF(2)). f : R → R defined by
f
((α βγ δ
))=
(α + γ 0
0 β + δ
)
for
(α βγ δ
)∈ R . Then f is a GF(2)-linear map.
1 [f (x), x6] = 0 for all x ∈ R .
2
[f
((0 10 0
)),
(0 10 0
)]=
(0 10 0
).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps
![Page 36: Decompositions of Quotient Rings and m-Commuting Maps](https://reader031.fdocuments.us/reader031/viewer/2022021007/62039483da24ad121e4b0ffe/html5/thumbnails/36.jpg)
Notations Decomposition of Q m-commuting maps Definitions Motivations Lemmas Main results Examples
In Case Q2 := M2(GF(2)).
Example 9 (T.-K. Lee, K.-S. Liu, W.-K. Shiue ,2004)
R := M2(GF(2)). f : R → R defined by
f
((α βγ δ
))=
(α + γ 0
0 β + δ
)
for
(α βγ δ
)∈ R . Then f is a GF(2)-linear map.
1 [f (x), x6] = 0 for all x ∈ R .
2
[f
((0 10 0
)),
(0 10 0
)]=
(0 10 0
).
Chih-Whi Chen Decompositions of Quotient Rings and m-Commuting Maps