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RINGS AS THE UNIONS OF PROPER SUBRINGS
Andrea Lucchini
Università di Padova, Italy
Una giornata per SilviaPadova, March 26 - 27, 2010
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXERCIZE
No group is the union of two of its proper subgroups.
SCORZA 1926
A group G is a union of three of its pairwise distinct proper subgroupsA, B, C if and only if
A, B, C have index 2 in G;
G/(A ∩ B ∩ C) is isomorphic to the Klein four group.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXERCIZE
No group is the union of two of its proper subgroups.
SCORZA 1926
A group G is a union of three of its pairwise distinct proper subgroupsA, B, C if and only if
A, B, C have index 2 in G;
G/(A ∩ B ∩ C) is isomorphic to the Klein four group.
QUESTIONS
Are there similar results for rings?
No ring is the union of two of its proper subrings.Is there an analogue of Scorza’s result for rings?
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
QUESTIONS
Is there an analogue of Scorza’s result for rings?
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
QUESTIONS
Is there an analogue of Scorza’s result for rings?
It is sufficient to classify all rings R and all proper subrings S1, S2, S3of R such that
R = S1 ∪ S2 ∪ S3
no non-trivial ideal of R is contained in S1 ∩ S2 ∩ S3.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
QUESTIONS
Is there an analogue of Scorza’s result for rings?
It is sufficient to classify all rings R and all proper subrings S1, S2, S3of R such that
R = S1 ∪ S2 ∪ S3
no non-trivial ideal of R is contained in S1 ∩ S2 ∩ S3.
DEFINITION
A 4-tuple (R, S1, S2, S3) of rings is good if S1, S2, S3 are propersubrings of the ring R so that R = S1 ∪ S2 ∪ S3 and that no non-trivialideal of R is contained in S1 ∩ S2 ∩ S3.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
QUESTIONS
Is there an analogue of Scorza’s result for rings?
It is sufficient to classify all rings R and all proper subrings S1, S2, S3of R such that
R = S1 ∪ S2 ∪ S3
no non-trivial ideal of R is contained in S1 ∩ S2 ∩ S3.
DEFINITION
A 4-tuple (R, S1, S2, S3) of rings is good if S1, S2, S3 are propersubrings of the ring R so that R = S1 ∪ S2 ∪ S3 and that no non-trivialideal of R is contained in S1 ∩ S2 ∩ S3.
THEOREM (ATTILA MAROTI - AL 2009)
All good 4-tuples of rings are completely described by the followingten Examples.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 1
R =
{(0 00 0
),
(1 00 1
),
(1 00 0
),
(0 00 1
)}≤ M2(Z/2Z).
A commutative ring of order 4 with a multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 1
R =
{(0 00 0
),
(1 00 1
),
(1 00 0
),
(0 00 1
)}≤ M2(Z/2Z).
A commutative ring of order 4 with a multiplicative identity.Every non-zero element lies inside a unique subring of order 2.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 1
R =
{(0 00 0
),
(1 00 1
),
(1 00 0
),
(0 00 1
)}≤ M2(Z/2Z).
A commutative ring of order 4 with a multiplicative identity.Every non-zero element lies inside a unique subring of order 2.R is the direct product of two fields of order 2.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 2
0 0 0
0 0 00 0 0
,
0 0 01 0 00 0 0
,
0 0 00 0 01 0 0
,
0 0 01 0 01 0 0
≤ M3(Z/2Z)
A commutative ring of order 4.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 2
0 0 0
0 0 00 0 0
,
0 0 01 0 00 0 0
,
0 0 00 0 01 0 0
,
0 0 01 0 01 0 0
≤ M3(Z/2Z)
A commutative ring of order 4.It has no multiplicative identity since it is a zero ring.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 2
0 0 0
0 0 00 0 0
,
0 0 01 0 00 0 0
,
0 0 00 0 01 0 0
,
0 0 01 0 01 0 0
≤ M3(Z/2Z)
A commutative ring of order 4.It has no multiplicative identity since it is a zero ring.Every non-zero element lies inside a unique subring of order 2.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLES 3 AND 4
R =
{(0 00 0
),
(0 10 0
),
(1 00 0
),
(1 10 0
)}≤ M2(Z/2Z).
Rop =
{(0 00 0
),
(0 01 0
),
(1 00 0
),
(1 01 0
)}≤ M2(Z/2Z).
Two non-commutative rings of order 4.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLES 3 AND 4
R =
{(0 00 0
),
(0 10 0
),
(1 00 0
),
(1 10 0
)}≤ M2(Z/2Z).
Rop =
{(0 00 0
),
(0 01 0
),
(1 00 0
),
(1 01 0
)}≤ M2(Z/2Z).
Two non-commutative rings of order 4.They have no multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLES 3 AND 4
R =
{(0 00 0
),
(0 10 0
),
(1 00 0
),
(1 10 0
)}≤ M2(Z/2Z).
Rop =
{(0 00 0
),
(0 01 0
),
(1 00 0
),
(1 01 0
)}≤ M2(Z/2Z).
Two non-commutative rings of order 4.They have no multiplicative identity.Every non-zero element lies inside a unique subring of order 2.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 5
R =
a 0 0
0 b 00 0 c
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 5
R =
a 0 0
0 b 00 0 c
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction a + b = 0S2 defined by the restriction a + c = 0S3 defined by the restriction b + c = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 5
R =
a 0 0
0 b 00 0 c
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction a + b = 0S2 defined by the restriction a + c = 0S3 defined by the restriction b + c = 0.
S1 ∩ S2 ∩ S3 =
a 0 0
0 a 00 0 a
∣∣∣a ∈ Z/2Z
is a subring of R, but
it is not an ideal.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 6
R =
a 0 0
b a 0c 0 a
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 6
R =
a 0 0
b a 0c 0 a
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction b = 0S2 defined by the restriction c = 0S3 defined by the restriction b + c = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 6
R =
a 0 0
b a 0c 0 a
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction b = 0S2 defined by the restriction c = 0S3 defined by the restriction b + c = 0.
The subset of R obtained by imposing the restriction a = 0 isisomorphic to the ring R of Example 2.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 6
R =
a 0 0
b a 0c 0 a
∣∣∣a, b, c ∈ Z/2Z
≤ M3(Z/2Z)
A commutative ring of order 8 with a multiplicative identity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction b = 0S2 defined by the restriction c = 0S3 defined by the restriction b + c = 0.
The subset of R obtained by imposing the restriction a = 0 isisomorphic to the ring R of Example 2.R can be obtained from Example 2 by adding a multiplicativeidentity 1 and imposing the relation 1 + 1 = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 7
R :=
{(a c0 b
) ∣∣∣a, b, c ∈ Z/2Z}≤ M2(Z/2Z)
A non-commutative ring of order 8 containing a multiplicativeidentity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 7
R :=
{(a c0 b
) ∣∣∣a, b, c ∈ Z/2Z}≤ M2(Z/2Z)
A non-commutative ring of order 8 containing a multiplicativeidentity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction c = 0S2 defined by the restriction a + b = 0S3 defined by the restriction a + b + c = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 7
R :=
{(a c0 b
) ∣∣∣a, b, c ∈ Z/2Z}≤ M2(Z/2Z)
A non-commutative ring of order 8 containing a multiplicativeidentity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction c = 0S2 defined by the restriction a + b = 0S3 defined by the restriction a + b + c = 0.
R ∼= Rop.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 7
R :=
{(a c0 b
) ∣∣∣a, b, c ∈ Z/2Z}≤ M2(Z/2Z)
A non-commutative ring of order 8 containing a multiplicativeidentity.R is the union of three subrings of order 4 containing themultiplicative identity:
S1 defined by the restriction c = 0S2 defined by the restriction a + b = 0S3 defined by the restriction a + b + c = 0.
R ∼= Rop.
R can be obtained from Examples 3 or 4 by adding amultiplicative identity 1 and imposing the relation 1 + 1 = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLES 8 AND 9
R =
0 b c d0 b 0 00 0 b 00 0 0 b
∣∣∣b, c, d ∈ Z/2Z
≤ M4(Z/2Z).
Rop =
0 0 0 0b b 0 0c 0 b 0d 0 0 b
∣∣∣b, c, d ∈ Z/2Z
≤ M4(Z/2Z).
Two non-commutative rings of order 8.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLES 8 AND 9
R =
0 b c d0 b 0 00 0 b 00 0 0 b
∣∣∣b, c, d ∈ Z/2Z
≤ M4(Z/2Z).
Rop =
0 0 0 0b b 0 0c 0 b 0d 0 0 b
∣∣∣b, c, d ∈ Z/2Z
≤ M4(Z/2Z).
Two non-commutative rings of order 8.They have no multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLES 8 AND 9
R =
0 b c d0 b 0 00 0 b 00 0 0 b
∣∣∣b, c, d ∈ Z/2Z
≤ M4(Z/2Z).
Rop =
0 0 0 0b b 0 0c 0 b 0d 0 0 b
∣∣∣b, c, d ∈ Z/2Z
≤ M4(Z/2Z).
Two non-commutative rings of order 8.They have no multiplicative identity.They are the union of three subrings of order 8:
S1 defined by the restriction c = 0S2 defined by the restriction d = 0S3 defined by the restriction c + d = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 10
R :=
a 0 0 0b e 0 0c 0 e 0d 0 0 e
∣∣∣a, b, c, d ∈ Z/2Z, a + b = e
≤ M2(Z/2Z)
A non-commutative ring of order 16 containing a multiplicativeidentity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 10
R :=
a 0 0 0b e 0 0c 0 e 0d 0 0 e
∣∣∣a, b, c, d ∈ Z/2Z, a + b = e
≤ M2(Z/2Z)
A non-commutative ring of order 16 containing a multiplicativeidentity.R is the union of three subring of order 8 containing themultiplicative identity:
S1 defined by the restriction c = 0S2 defined by the restriction d = 0S3 defined by the restriction c + d = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 10
R :=
a 0 0 0b e 0 0c 0 e 0d 0 0 e
∣∣∣a, b, c, d ∈ Z/2Z, a + b = e
≤ M2(Z/2Z)
A non-commutative ring of order 16 containing a multiplicativeidentity.R is the union of three subring of order 8 containing themultiplicative identity:
S1 defined by the restriction c = 0S2 defined by the restriction d = 0S3 defined by the restriction c + d = 0.
R ∼= Rop.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 10
R :=
a 0 0 0b e 0 0c 0 e 0d 0 0 e
∣∣∣a, b, c, d ∈ Z/2Z, a + b = e
≤ M2(Z/2Z)
A non-commutative ring of order 16 containing a multiplicativeidentity.R is the union of three subring of order 8 containing themultiplicative identity:
S1 defined by the restriction c = 0S2 defined by the restriction d = 0S3 defined by the restriction c + d = 0.
R ∼= Rop.
R can be obtained from Examples 8 or 9 by adding amultiplicative identity 1 and imposing the relation 1 + 1 = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
EXAMPLE 10
R :=
a 0 0 0b e 0 0c 0 e 0d 0 0 e
∣∣∣a, b, c, d ∈ Z/2Z, a + b = e
≤ M2(Z/2Z)
(R, +) = 〈1, x , y , z〉 ∼= (Z/2Z)4 andz2 = z,x2 = y2 = xy = yx = 0,xz = x , yz = y , zx = 0, zy = 0.
S1 = 〈1, x , z〉, S2 = 〈1, y , z〉, S3 = 〈1, x + y , z〉
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
In same of the previous examples, R contains and ideal I such thatR/I is isomorphic to one of the other examples.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
In same of the previous examples, R contains and ideal I such thatR/I is isomorphic to one of the other examples.
Examples 5, 7, 10 have Example 1 as an epimorphic image.Example 8 has Example 4 as an epimorphic image.Example 9 has Example 3 as an epimorphic image.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
In same of the previous examples, R contains and ideal I such thatR/I is isomorphic to one of the other examples.
Examples 5, 7, 10 have Example 1 as an epimorphic image.Example 8 has Example 4 as an epimorphic image.Example 9 has Example 3 as an epimorphic image.
But this is not the case of Example 6.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
In same of the previous examples, R contains and ideal I such thatR/I is isomorphic to one of the other examples.
Examples 5, 7, 10 have Example 1 as an epimorphic image.Example 8 has Example 4 as an epimorphic image.Example 9 has Example 3 as an epimorphic image.
But this is not the case of Example 6.
THEOREM
A ring R is the union of three of its proper subrings if and only if thereexists a factor ring (of order 4 or 8) of R which is isomorphic to ring ofexamples 1, 2, 3, 4 or 6.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
HOW CAN WE PROVE OUR RESULT?
Let (R, S1, S2, S3) be a good 4-tuple of rings and let S = S1∩S2∩S3.Scorza’s Theorem⇒ |R : Si | = 2 for each i ∈ {1, 2, 3} andS1 ∩ S2 = S1 ∩ S3 = S2 ∩ S3 = S.2R ⊆ S1 ∩ S2 ∩ S3 ⇒ 2R = 0.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
HOW CAN WE PROVE OUR RESULT?
Let (R, S1, S2, S3) be a good 4-tuple of rings and let S = S1∩S2∩S3.Scorza’s Theorem⇒ |R : Si | = 2 for each i ∈ {1, 2, 3} andS1 ∩ S2 = S1 ∩ S3 = S2 ∩ S3 = S.2R ⊆ S1 ∩ S2 ∩ S3 ⇒ 2R = 0.
S is a subring of R with |R : S| = 4.
[J. Lewin (1967)] Suppose that a ring R contains a subring S offinite index. Then there exists an ideal I of R, contained in S andof finite index in R.R is finite (|R| ≤ 525).
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
A USEFUL REMARK
Let M be a ring (with or without a multiplicative identity) with2M = 0.
Consider the abelian group M∗ = M ⊕ 〈u〉 with u + u = 0 anddefine a multiplication on M∗ by setting u to be the identity on M∗.
M∗ becomes a ring with a multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
A USEFUL REMARK
Let M be a ring (with or without a multiplicative identity) with2M = 0.
Consider the abelian group M∗ = M ⊕ 〈u〉 with u + u = 0 anddefine a multiplication on M∗ by setting u to be the identity on M∗.
M∗ becomes a ring with a multiplicative identity.
Let (R, S1, S2, S3) be a good 4-tuple . Suppose that R has nomultiplicative identity. Then (R∗, S∗
1 , S∗2 , S∗
3 ) is also a good 4-tuplewhere a unique multiplicative identity was added to the four rings R,S1, S2, and S3.
We may restrict our attention to rings with a multiplicative identity.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
sx ∈ S, sy /∈ S
⇓
sx = s1 ∃ s1 ∈ S, sy = s2 + y ∃ s2 ∈ S
⇓
s(x + y) = s1 + s2 + y /∈ S3
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
SR := {s ∈ S | sx ∈ S}, SL := {s ∈ S | xs ∈ S}, T := SL ∩ SR .
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
SR := {s ∈ S | sx ∈ S}, SL := {s ∈ S | xs ∈ S}, T := SL ∩ SR .SR and ST are subgroups of S with index at most 2, so T is asubgroup of S with |S : T | ∈ {1, 2, 4}.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
SR := {s ∈ S | sx ∈ S}, SL := {s ∈ S | xs ∈ S}, T := SL ∩ SR .SR and ST are subgroups of S with index at most 2, so T is asubgroup of S with |S : T | ∈ {1, 2, 4}.t ∈ T ⇒ xty , ytx , xtx , yty ∈ S
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
SR := {s ∈ S | sx ∈ S}, SL := {s ∈ S | xs ∈ S}, T := SL ∩ SR .SR and ST are subgroups of S with index at most 2, so T is asubgroup of S with |S : T | ∈ {1, 2, 4}.t ∈ T ⇒ xty , ytx , xtx , yty ∈ S{
xt ∈ S ⇒ xty ∈ S2
ty ∈ S ⇒ xty ∈ S1⇒ xty ∈ S1 ∩ S2 = S
{xtx = s1 + bxxty = s2
⇒ xt(x +y) = s1+s2+bx ∈ S3 b = 0
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let R be a good ring. Then |R| = 4, 8, or 16.
PROOF
There exists x and y such that R = S ⊕ {x , y , x + y , 0},S1 = S ⊕ {x , 0}, S2 = S ⊕ {y , 0}, S3 = S ⊕ {x + y , 0}.∀s ∈ S we have: sx ∈ S ⇔ sy ∈ S, xs ∈ S ⇔ ys ∈ S,
SR := {s ∈ S | sx ∈ S}, SL := {s ∈ S | xs ∈ S}, T := SL ∩ SR .SR and ST are subgroups of S with index at most 2, so T is asubgroup of S with |S : T | ∈ {1, 2, 4}.t ∈ T ⇒ xty , ytx , xtx , yty ∈ SRTR ⊆ S ⇒ RTR = 0⇒ T = 0⇒ |S| ∈ {1, 2, 4}.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let (R, S1, S2, S3) be a good 4-tuple of rings. If R contains amultiplicative identity 1, then either |R| = 4 or 1 ∈ S1 ∩ S2 ∩ S3.
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
PROPOSITION
Let (R, S1, S2, S3) be a good 4-tuple of rings. If R contains amultiplicative identity 1, then either |R| = 4 or 1 ∈ S1 ∩ S2 ∩ S3.
PROOF
1 /∈ S
⇓
we may assume 1 /∈ S1, 1 /∈ S2
⇓
S1 and S2 are ideals of R
⇓
S = S1 ∩ S2 is an ideal of R
⇓
S = {0}ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
Thanks
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
Thanks
to all the participants
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
Thanks
to all the participants
to the organizers (in particular to Eloisa and Alberto)
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
Thanks
to all the participants
to the organizers (in particular to Eloisa and Alberto)
but especially
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
IS THERE AN ANALOGUE OF SCORZA’S THEOREM FOR RINGS?EXAMPLES
PROOFS
Thanks
to all the participants
to the organizers (in particular to Eloisa and Alberto)
but especially
thanks to Silvia, for the time she spent with us
ANDREA LUCCHINI RINGS AS THE UNIONS OF PROPER SUBRINGS
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