Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local...

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Understanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Transcript of Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local...

Page 1: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Understanding when Finite Local Rings shareZero Divisor Graphs

Sarah Fletcher

MSU REU

30 July 2008

Page 2: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Zero Divisor GraphLet R be a ring and let Z∗(R) denote the set of zero divisorsof R.

Definition

Γ(R) = (Z∗(R), {(a, b)|a, b ∈ Z∗(R), ab = 0})

Example:R = Z12

2

34

6

89

10

Page 3: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Zero Divisor GraphLet R be a ring and let Z∗(R) denote the set of zero divisorsof R.

Definition

Γ(R) = (Z∗(R), {(a, b)|a, b ∈ Z∗(R), ab = 0})

Example:R = Z12

2

34

6

89

10

Page 4: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Zero Divisor GraphLet R be a ring and let Z∗(R) denote the set of zero divisorsof R.

Definition

Γ(R) = (Z∗(R), {(a, b)|a, b ∈ Z∗(R), ab = 0})

Example:R = Z12

2

34

6

89

10

Page 5: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Local Rings of Order 16, |M| = 8

• Characteristic 2: Z2[x]/(x4), Z2[x, y]/(x3, xy, y2),Z2[x, y]/(x3, xy, x2 − y2), Z2[x, y]/(x2, y2),andZ2[x, y, z]/(x, y, z)2

• Characteristic 4: Z4[x]/(x2, 2x), Z4[x]/(x2), Z4[x]/(x2 − 2x),Z4[x, y]/(x3, y2, xy, x2 − 2),Z4[x, y]/(x3, x2 − 2, x2 − y2, xy),and Z[x, y]/(x2, y2, xy − 2)

• Characteristic 8: Z8[x]/(x3, x2 − 4, 2x), andZ8[x]/(x3, x2 − 4, 2x)

• Characteristic 16: Z16

Page 6: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

x x2

x2+ x

x3

x3+ x

x3+ x2

x3+ x2

+ x

Z2@xD�Hx4L

24

6

8

10

1214

Z16

Page 7: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

x x2

x2+ x

x3

x3+ x

x3+ x2

x3+ x2

+ x

Z2@xD�Hx4L

24

6

8

10

1214

Z16

Page 8: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

• Z2[x, y]/(x3, xy, y2)

• Z4[x, y]/(x3, x2 − 2, xy, y2)

• Z4[x]/(x3, 2x)

• Z8[x]/(x2, 2x)

Page 9: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

• Z2[x, y]/(x3, xy, x2 − y2)

• Z4[x, y]/(x3, xy, x2 − 2, x2 − y2)

• Z4[x]/(x2 − 2x)

• Z8[x]/(x3, x2 − 4, 2x)

Page 10: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

• Z2[x, y]x2, y2)

• Z4[x, y]/(x2, y2, xy − 2)

• Z4[x]/(x2)

Page 11: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

• Z2[x, y, z]/(x, y, z)2

• Z16

Page 12: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Local Rings of Order 16, |M| = 8

Characteristic 2

Z2[x]/(x4)

Z2[x, y]/(x3, xy, y2)

Z2[x, y]/(x3, xy, x2 − y2)

Z2[x, y]/(x2, y2)

Z2[x, y, z]/(x, y, z)2

Page 13: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Galois Rings

Let τ∗ be the unique cyclic multiplicative subgroup of order pr − 1in the Galois ring

G(pk, r) = Zpk [x]/(f).

The Teichmuller set is τ = τ∗ ∪ {0}.Then any element a ∈ G(pk, r) can be uniquely represented as

a = a0 + a1p+ a2p2 + · · ·+ ak−1p

k−1.

Page 14: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Galois Rings

Let τ∗ be the unique cyclic multiplicative subgroup of order pr − 1in the Galois ring

G(pk, r) = Zpk [x]/(f).

The Teichmuller set is τ = τ∗ ∪ {0}.Then any element a ∈ G(pk, r) can be uniquely represented as

a = a0 + a1p+ a2p2 + · · ·+ ak−1p

k−1.

Page 15: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Define ϕ : G(pk, r)→ Fpr [y]/(yk) so that

a 7→ π(a0) + π(a1)y + π(a2)y2 + · · ·+ π(ak−1)yk−1

where the ai come from the Teichmuller representation of a.

Theoremϕ induces a graph isomorphism between Γ(G(pk, r)) andΓ(Fpr [y]/(yk)).

Page 16: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Define ϕ : G(pk, r)→ Fpr [y]/(yk) so that

a 7→ π(a0) + π(a1)y + π(a2)y2 + · · ·+ π(ak−1)yk−1

where the ai come from the Teichmuller representation of a.

Theoremϕ induces a graph isomorphism between Γ(G(pk, r)) andΓ(Fpr [y]/(yk)).

Page 17: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

ϕ(a) = π(a0) + π(a1)y + π(a2)y2 + · · ·+ π(ak−1)y

k−1

Example: G(24, 1) = Z16

Note that τ = {0, 1}.

24

6

8

10

1214

Z16

x x2

x2+ x

x3

x3+ x

x3+ x2

x3+ x2

+ x

Z2@xD�Hx4L

Page 18: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

ϕ(a) = π(a0) + π(a1)y + π(a2)y2 + · · ·+ π(ak−1)y

k−1

Example: G(24, 1) = Z16

Note that τ = {0, 1}.

24

6

8

10

1214

Z16

x x2

x2+ x

x3

x3+ x

x3+ x2

x3+ x2

+ x

Z2@xD�Hx4L

Page 19: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

ϕ(a) = π(a0) + π(a1)y + π(a2)y2 + · · ·+ π(ak−1)y

k−1

Example: G(24, 1) = Z16

Note that τ = {0, 1}.

24

6

8

10

1214

Z16

x x2

x2+ x

x3

x3+ x

x3+ x2

x3+ x2

+ x

Z2@xD�Hx4L

Page 20: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Finite Local Rings

Let G(pk, r)[x1, . . . , xn]/I be a finite local ring and let

R = G(pk, r)[x1, . . . , xn]

S = Fpr [y]/yk[x1, . . . , xn].

Define ψ : R→ S by ∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Page 21: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Finite Local Rings

Let G(pk, r)[x1, . . . , xn]/I be a finite local ring and let

R = G(pk, r)[x1, . . . , xn]

S = Fpr [y]/yk[x1, . . . , xn].

Define ψ : R→ S by ∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Page 22: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Finite Local Rings

Let G(pk, r)[x1, . . . , xn]/I be a finite local ring and let

R = G(pk, r)[x1, . . . , xn]

S = Fpr [y]/yk[x1, . . . , xn].

Define ψ : R→ S by ∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Page 23: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Finite Local Rings

Let G(pk, r)[x1, . . . , xn]/I be a finite local ring and let

R = G(pk, r)[x1, . . . , xn]

S = Fpr [y]/yk[x1, . . . , xn].

Define ψ : R→ S by ∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Page 24: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Making Things Finite Again

Given the finite local ring G(pk, r)[x1, . . . , xn]/I, let

Q = ({xmi |xmi ∈ I})

RQ = G(pk, r)[x1, . . . , xn]/Q SQ = Fpr [y]/yk[x1, . . . , xn]/Q.

Define ψQ : RQ → SQ by∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Note that we can find an I ′ so that

RQ/I′ ∼= R/I.

Page 25: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Making Things Finite Again

Given the finite local ring G(pk, r)[x1, . . . , xn]/I, let

Q = ({xmi |xmi ∈ I})

RQ = G(pk, r)[x1, . . . , xn]/Q SQ = Fpr [y]/yk[x1, . . . , xn]/Q.

Define ψQ : RQ → SQ by∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Note that we can find an I ′ so that

RQ/I′ ∼= R/I.

Page 26: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Making Things Finite Again

Given the finite local ring G(pk, r)[x1, . . . , xn]/I, let

Q = ({xmi |xmi ∈ I})

RQ = G(pk, r)[x1, . . . , xn]/Q SQ = Fpr [y]/yk[x1, . . . , xn]/Q.

Define ψQ : RQ → SQ by∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Note that we can find an I ′ so that

RQ/I′ ∼= R/I.

Page 27: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

Making Things Finite Again

Given the finite local ring G(pk, r)[x1, . . . , xn]/I, let

Q = ({xmi |xmi ∈ I})

RQ = G(pk, r)[x1, . . . , xn]/Q SQ = Fpr [y]/yk[x1, . . . , xn]/Q.

Define ψQ : RQ → SQ by∑α∈Nn

aαxα 7→ ϕ(aα)xα.

Note that we can find an I ′ so that

RQ/I′ ∼= R/I.

Page 28: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

An Idea

J = {ψQ(a)ψQ(b)|ab ∈ I ′} ⊆ SQ

Compare:

Γ(RQ/I′) and Γ(SQ/(J)).

Page 29: Understanding when Finite Local Rings share Zero Divisor ... fileUnderstanding when Finite Local Rings share Zero Divisor Graphs Sarah Fletcher MSU REU 30 July 2008

An Idea

J = {ψQ(a)ψQ(b)|ab ∈ I ′} ⊆ SQ

Compare:

Γ(RQ/I′) and Γ(SQ/(J)).