Decompositions of Quotient Rings and m-Commuting Maps

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Notations Decomposition ofQ m-commuting maps

Decompositions of Quotient Rings and m-Commuting Maps

Chih-Whi Chen

Department of Mathematics National Taiwan University

July 14th, 2011

Chih-Whi Chen Decompositions of Quotient Rings andm-Commuting Maps

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Notations Decomposition ofQ m-commuting maps Semiprimeness Polynomial identities

R : semiprime ring.

Q : symmetric Martindale quotient ring of R. C : extended centroid of R.

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R : semiprime ring.

Q : symmetric Martindale quotient ring of R.

C : extended centroid of R.

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R : semiprime ring.

Q : symmetric Martindale quotient ring of R. C : extended centroid of R.

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Z{Xˆ}the free Z-algebra in noncommutative indeterminates in ˆX :={X,X₁,X₂, . . .}.

R satisfies f(X₁, . . . ,X_t)∈Z{Xˆ} if

f(r₁, . . . ,r_t) = 0 for all r_i ∈R.

R is called faithful f-free if I doesn’t satisfy f for every nonzeroI CR.

S_k(X₁, . . . ,X_k) := P

σ∈Sym(k)

(−1)^σX_σ(1)· · ·X_σ(k). For example,S2(X1,X2) =X1X2−X2X1.

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f(r₁, . . . ,r_t) = 0 for all r_i ∈R.

R is called faithful f-free if I doesn’t satisfy f for every nonzeroI CR.

S_k(X₁, . . . ,X_k) := P

σ∈Sym(k)

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f(r₁, . . . ,r_t) = 0 for all r_i ∈R.

R is called faithful f-free if I doesn’t satisfyf for every nonzeroI CR.

S_k(X₁, . . . ,X_k) := P

σ∈Sym(k)

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f(r₁, . . . ,r_t) = 0 for all r_i ∈R.

R is called faithful f-free if I doesn’t satisfyf for every nonzeroI CR.

S_k(X₁, . . . ,X_k) := P

σ∈Sym(k)

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Notations Decomposition ofQ m-commuting maps Motivations Main results

Theorem 1 (T. Kosan, T.-K. Lee, Y. Zhou)

Let f(X)∈Z{Xˆ}. 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.

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Theorem 2 (C. -W. Chen and T. -K. Lee, 2011)

Let n≥2 and f(X) = Xⁿh(X) where h(X)is a polynomial overZ 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 ∼=M_n(E) where E is a commutative self-injective ring such that, for some fixed integer q>1, x^q =x for all x ∈E .

3 Q₃ is a both faithful S_2n−2-free and faithful f -free ring.

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Notations Decomposition ofQ 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),x^m] = 0 for all x ∈R.

Definition 4

D(R) := {δ∈Der(Q)|I^δ ⊆I,for some I C^ess.R}. A derivaition word: ∆ =δ₁δ₂· · ·δ_t, each δ_k ∈D(R). A map by linear generalized differential polynomial:

X →X

i

X

j

a_ijX^∆^jb_ij

wherea_ij,b_ij ∈Q and derivation words ∆_j.

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Definition 3

Definition 4

D(R) := {δ∈Der(Q)|I^δ ⊆I,for some I C^ess.R}.

A derivaition word: ∆ =δ₁δ₂· · ·δ_t, each δ_k ∈D(R). A map by linear generalized differential polynomial:

X →X

i

X

j

a_ijX^∆^jb_ij

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Definition 3

Definition 4

A derivaition word: ∆ =δ₁δ₂· · ·δ_t, each δ_k ∈D(R).

A map by linear generalized differential polynomial:

X →X

i

X

j

a_ijX^∆^jb_ij

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Definition 3

Definition 4

A derivaition word: ∆ =δ₁δ₂· · ·δ_t, each δ_k ∈D(R).

A map by linear generalized differential polynomial:

X →X

i

X

j

a_ijX^∆^jb_ij

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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),x^m] = 0 for all x ∈R, where ψ defined by a linear generalized differential polynomial. Then

ψ(x) = λx +µ(x) for all x ∈R, where λ∈C and µ:R →C .

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Lemma 6 (C.-W. Chen and T.-K. Lee, 2011) Let

f(X) =X²(X²+ 1)(X²+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 is commutative, nor Q/P ∼=M₂(GF(2)).

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f(X) =X²(X²+ 1)(X²+X + 1).

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f(X) =X²(X²+ 1)(X²+X + 1).

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Theorem 7 (C.-W. Chen and T.-K. Lee, 2011) Let

f(X) =X²(X²+ 1)(X²+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.

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f(X) =X²(X²+ 1)(X²+X + 1).

1 Q1 is a commutative ring.

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f(X) =X²(X²+ 1)(X²+X + 1).

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f(X) =X²(X²+ 1)(X²+X + 1).

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Theorem 8 (C.-W. Chen and T.-K. Lee, 2011)

Suppose thatψ :Q →Q satisfying [ψ(x),x^m] = 0 for all x ∈R, where ψ defined by a linear generalized differential polynomial. Then

2 Q₂ ∼=M₂(E) where E is a complete Boolean algebra.

3 ψ(x) = λx +µ(x) for all x ∈Q₃, where λ∈C and µ:Q3 →C .

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In Case Q

₂

:= M

₂

(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 β

γ δ

=

α+γ 0 0 β+δ

for

α β

γ δ

∈R. Then f is a GF(2)-linear map.

1 [f(x),x⁶] = 0 for all x ∈R.

2

f

0 1 0 0

,

0 1 0 0

=

0 1 0 0

.

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In Case Q

₂

:= M

₂

(GF(2)).

f

α β

γ δ

=

α+γ 0 0 β+δ

for

α β

γ δ

∈R.

Then f is a GF(2)-linear map.

1 [f(x),x⁶] = 0 for all x ∈R.

2

f

0 1 0 0

,

0 1 0 0

=

0 1 0 0

.

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In Case Q

₂

:= M

₂

(GF(2)).

f

α β

γ δ

=

α+γ 0 0 β+δ

for

α β

γ δ

1 [f(x),x⁶] = 0 for all x ∈R.

2

f

0 1 0 0

,

0 1 0 0

=

0 1 0 0

.

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In Case Q

₂

:= M

₂

(GF(2)).

f

α β

γ δ

=

α+γ 0 0 β+δ

for

α β

γ δ

1 [f(x),x⁶] = 0 for all x ∈R.

2

f

0 1 0 0

,

0 1 0 0

=

0 1 0 0

.

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In Case Q

₂

:= M

₂

(GF(2)).

f

α β

γ δ

=

α+γ 0 0 β+δ

for

α β

γ δ

1 [f(x),x⁶] = 0 for all x ∈R.

f

0 1 ,

0 1

=

0 1 .