Download Derivations in semiprime rings

Derivations of a ring and simplicity
Derivations in semiprime rings
Orest Artemovych
Cracow University of Technology
Cracow, Poland
Groups and Their Actions
Conference in Be˛dlewo
22 Juny 2015
Derivations of a ring and simplicity
Derivations of a ring and simplicity
Introduction.
An Abelian group (D, +) with an algebraic operation [−, −] is
called a Lie ring if, for all elements x, y , z ∈ D, the following hold:
1) (anticommutativity)
[x, x] = 0,
2) (linearity)
• [x + y , z] = [x, z] + [y , z],
• [x, y + z] = [x, y ] + [x, z],
3) (the Jacobi identity)
[x, [y , z]] + [y , [z, x]] + [z, [x, y ]] = 0.
Derivations of a ring and simplicity
Definition of an associative ring
In the next
(R, +, ·)
is an associative ring with the identity 1, i.e.
• (R, +) is an Abelian group (with the zero element 0),
• the multiplication · is associative,
• the multiplication · is left and right distributive with respect to
the addition +,
• ∃1∈R ∀a∈R : 1 · a = a = a · 1.
Derivations of a ring and simplicity
Definition of a derivation
A map d : R → R is called a derivation of a ring R if
d(a + b) = d(a) + d(b)
and the Leibnitz rule holds
d(ab) = d(a)b + ad(b)
for all a, b ∈ R.
Derivations of a ring and simplicity
The zero derivation
A zero map
0R : R 3 r 7→ 0 ∈ R
is a derivation of a ring R.
Derivations of a ring and simplicity
Lie ring of derivations We denote by
DerR = {d : R → R | d is a derivation}
the set of all derivations in R.
Derivations of a ring and simplicity
Lemma (Jacobson, 1937).
———————————————————————————–
(DerR, +, [−, −]) is a Lie ring under two operations:
(i) (pointwise addition + of derivations)
(d1 + d2 )(r ) = d1 (r ) + d2 (r ),
(ii) (pointwise Lie multiplication [−, −] of derivations)
[d1 , d2 ](r ) = d1 (d2 (r )) − d2 (d1 (r ))
for all d1 , d2 ∈ DerR and r ∈ R.
Derivations of a ring and simplicity
Definition of an inner derivation
The rule
∂x : R 3 r 7→ xr − rx ∈ R
determines a derivation of R (so-called an inner derivation of R
generated by x ∈ R).
∗∗∗
An element [x, r ] = xr − rx is also called the (additive)
commutator of elements x, r ∈ R.
∗∗∗
The zero map 0R = ∂0 is an inner derivation.
Derivations of a ring and simplicity
Lie ideal
We denote by
IDerR = {∂x : R → R | x ∈ R}
the set of all inner derivations in R.
Derivations of a ring and simplicity
Recall that a subset B of a Lie ring (D, +, [−, −]) is called a (Lie)
ideal of D if
• (B, +) is a subgroup of (D, +),
• [b, d] ∈ B for all b ∈ B and d ∈ D.
Derivations of a ring and simplicity
Lemma (Jacobson, 1937).
———————————————————————————–
IDerR is a (Lie) ideal in DerR.
Derivations of a ring and simplicity
Simple, prime, semisimple
A (Lie or associative) ring A is said to be:
• simple if there no two-sided ideals other 0 or A,
• prime if, for all ideals K , S of A, the condition KS = 0 implies
that K = 0 or S = 0,
• semiprime if, for any ideal K of A, the condition K 2 = 0
implies that K = 0,
• primary if, for any ideals K , S of A, the condition KS = 0
implies that K = 0 or S is nilpotent.
Derivations of a ring and simplicity
In the next
————
• (the commutator set)
[R, R] = {[r , t] | r , t ∈ R}.
• (the commutator ideal)
C (R) is the commutator ideal, i.e. the ideal of R generated by
all [r , t], where r , t ∈ R.
Derivations of a ring and simplicity
Some aspects of simplicity
Some aspects of simplicity:
A.
E. Noether (193?) and (independently) T. Skolem (1927) prove the
following
——————————————————————————————–
Theorem.
IDerR = DerR for any finite dimensional algebra R over its center
Z (R).
N. Jacobson (1937) has proved that any derivation of a simple
algebra with an identity over an algebraic closed field of
characteristic zero is inner.
∗∗∗
Derivations of a ring and simplicity
Open problem. Characterize simple rings (in particular infinite
dimensional skew fields) R with all derivations to be inner:
IDerR = DerR.
Derivations of a ring and simplicity
B.
I.N. Herstein (1955–1961) proved
——————————————————————Proposition.
(1) If R is simple and not a field, then [R, R] = R.
(2) (see also S. Amitsur, 1957) If R is simple, then
[[R, R], [R, R]] = [R, R].
Derivations of a ring and simplicity
Definition of differentially trivial rings
For the next we need
———————————Definition.
If Der R = {0R }, then R is called differentially trivial.
∗∗∗
Every differentially trivial ring is commutative.
Derivations of a ring and simplicity
D.
We prove
———————————————————
Theorem.
Let R be a semiprime ring. If IDer R = Der R is simple Lie ring,
then R is a simple ring or a differentially trivial ring.
Derivations of a ring and simplicity
We prove:
Theorem.
Let R be a ring with an identity. If Der R = IDer R is a simple Lie
ring, then
R = C (R) ⊕ K
is a ring direct sum of a differentially trivial ring K and a simple
ring C (R) with only inner derivations, where C (R) is the
commutator ideal of R.
Derivations of a ring and simplicity
F.
Let d ∈ DerR and I be an ideal of R.
Then I is called d-ideal (or a differential ideal) if d(I ) ⊆ I .
If ∆ ⊆ DerR and d(I ) ⊆ I for all d ∈ ∆, then I is called ∆-ideal.
∗∗∗
E.C. Posner (1960) introduced the definition: a ring R is called
differentiably simple (or d-simple) if it contains only trivial
d-ideals (i.e., {0} and R) for some d ∈ DerR.
Derivations of a ring and simplicity
L. Harper (1961) (see also V. Bavula (2008)) proved
——————————————————————————————–
Theorem.
A Noetherian commutative ring R of characteristic p > 0 is a
differentiably simple ring if and only if it has the form
R = k[X1 , . . . , Xn ]/(X1p , . . . , Xnp ),
where k is a field of characteristic p.
Derivations of a ring and simplicity
R.E. Block (1969) proved
Theorem.
If R is a differentiably simple ring with a minimal ideal, then either
R is simple or there exist a simple ring S of prime characteristic
p > 0 and positive integer n such that
R = S[G ]
is a group ring of a non-trivial finite elementary abelian p-group G
over a ring S (this n is a number of direct factors of G ).
∗∗∗
Open problem. Characterize differentiably simple rings R (with
the identity 1) without minimal ideals.
∗∗∗
Open problem. Characterize differentiably simple nil-rings.
Derivations of a ring and simplicity
G.
D.A. Jordan, C.R. Jordan (1978–2000), A. Nowicki (1985) and
D.S. Passman (1998) study the Lie ring of derivations of a
D-simple commutative rings.
∗∗∗
D. Jordan (2000) proved the following
———————————————————————–
Theorem.
Let R be a commutative ring, D a non-zero Lie subring and a
submodule of Der R. If R is a D-simple, then:
(1) D is Lie simple except possibly when charR = 2 and D is
cyclic as an R-module.
(2) If charR = 2 and D = Rd is cyclic as an R-module, then D is
Lie simple if and only if d(R) = R.
Derivations of a ring and simplicity
In non-commutative case we (with Ahmad al Khalaf) prove
Theorem.
If R is a D-simple ring of characteristic 6= 2, then one of the
following holds:
(1) D is a simple Lie ring,
(2) R is a commutative ring,
(2) R is a simple ring.
Derivations of a ring and simplicity
On the set R we consider the Lie multiplication “[−, −]" defined by
the rule
[a, b] = a · b − b · a
for any a, b ∈ R. Then
R L = (R, +, [−, −])
is a Lie ring
Derivations of a ring and simplicity
We prove (with Maria Lukashenko, 2015) the following
Proposition.
Let R be a ring. Then the following statements hold:
(1) IDer R is a simple Lie ring if and only if R L /Z (R) is a simple
Lie ring,
(2) IDer R is a prime Lie ring if and only if R L /Z (R) is a prime Lie
ring,
(3) IDer R is a semiprime Lie ring if and only if R L /Z (R) is a
semiprime Lie ring,
(4) IDer R is a primary Lie ring if and only if R L /Z (R) is a
primary Lie ring.
Derivations of a ring and simplicity
M. A. Chebotar and P.-H. Lee (2006) have shown that if R is a
reduced (i.e., without nonzero nilpotent elements) 2-torsion free
commutative D-prime ring, D is a nonzero Lie subring and an
R-submodule of Der(R), δ ∈ Der(R), then D is a prime Lie ring.
∗∗∗
P.-H. Lee and C.-K. Liu (2007) have establish that if R is a
2-torsion free commutative D-prime ring, D is a Lie subring and an
R-submodule of Der(R), then any nonzero Lie ideal A of D is a
prime Lie ring.
Derivations of a ring and simplicity
We (with Ahmad al Khalaf, 2015) prove the following
Proposition.
Let R be a ring. If D = Der R is a prime Lie ring, then on of the
following holds:
(1) R is a commutative ring,
(2) R is a D-prime Lie ring.
Derivations of a ring and simplicity
Bibliography
I
N. Jacobson, Abstract derivation and Lie algebras, Trans.
Amer. Math. Soc. 42(1937), no. 2, 206–224.
I
I.N. Herstein, The Lie ring of a simple, associative ring, Duke
Math. J. 22(1955), 471–476.
I
I.N. Herstein, Lie and Jordan structures in simple, associative
rings, Bull. Amer. Math. Soc. 67(1961), 517–531.
Derivations of a ring and simplicity
I
S. Amitsur, Derivations in simple rings, Proc. London Math.
Soc. (3)7(1957), 87–112.
I
M. Chebotar, P.-H. Lee and E.R. Puczylowski, On
commutators and nilpotent elements in simple rings, Bull.
London Math. Soc. 42(2010), 191–194.
I
E.C. Posner, Differentiably simple rings, ???????
I
L. Harper, On differentiably simple algebras, Trans. Amer.
Math. Soc. 100(1961), 63–72.
Derivations of a ring and simplicity
I
R.E. Block, Differentiably simple rings?????? 1969.
I
V.V. Bavula, Simple derivations of differentiably simple
Noetherian commutative rings in prime characteristic, ??????
2008.
I
C.R. Jordan and D.A. Jordan, Lie rings of derivations of
associative rings, J. London Math. Soc. (2)17(1978), 33–41.
I
C.R. Jordan and D.A. Jordan, The Lie structure of a
commutative ring with a derivation, J. London Math. Soc.
(2)18(1978), 39–49.
I
D.A. Jordan, Simple Lie rings of derivations of commutative
rings, J. London Math. Soc. (2)18(1978), 443–448.
Derivations of a ring and simplicity
I
D.A. Jordan, On the ideals od a Lie algebra of derivations, J.
London Math. Soc. (2)33(1986), 443–448.
I
D.A. Jordan, On the ideals od a Lie algebra of derivations, J.
London Math. Soc. (2)33(1986), 443–448.
I
D.A. Jordan, On the simplicity of Lie algebras of derivations of
commutative algebras, J. Algebra 228(2000), 580–585.
I
D.S. Passman, Simple Lie algebras of Witt type, J. Algebra
34(1998), 682–692.
Derivations of a ring and simplicity
I
A. Nowicki, The Lie structure of a commutative ring with a
derivation, Archiv Math. (Basel) 45(1985), 33–39.
I
O.D. Artemovych, Differentially trivial and rigid rings of finite
rank, Periodica Math. Hungarica (1-2)36(1998), 1–16.
I
O.D. Artemovych, Differentially trivial left Noetherian rings,
Comment. Math. Univ. Carolinae (2)40(1999), 201–208.
Derivations of a ring and simplicity
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