Noetherian rings whose injective hulls of simple modules are locally Artinian.

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1 Noetherian rings whose injective hulls of simple modules are locally Artinian. Christian Lomp joint work with Paula Carvalho and Can Hatipoğlu Universidade do Porto Christian Lomp Locally Artinian injective hulls of simples. 1/14

2 ( ) Let R be an associative unital ring. Definition ( ) Injective hulls of simple left R-modules are locally Artinian. Christian Lomp Locally Artinian injective hulls of simples. 2/14

3 ( ) Let R be an associative unital ring. Definition ( ) Injective hulls of simple left R-modules are locally Artinian. ( ) any finitely generated essential extension of a simple left R-module is Artinian. Christian Lomp Locally Artinian injective hulls of simples. 2/14

4 ( ) Let R be an associative unital ring. Definition ( ) Injective hulls of simple left R-modules are locally Artinian. ( ) any finitely generated essential extension of a simple left R-module is Artinian. Pathological cases are Artinian rings and V-rings. Christian Lomp Locally Artinian injective hulls of simples. 2/14

5 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); Christian Lomp Locally Artinian injective hulls of simples. 3/14

6 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); 2 FBN rings (Jategaonkar); Christian Lomp Locally Artinian injective hulls of simples. 3/14

7 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); 2 FBN rings (Jategaonkar); ( ) Jacobson Conjecture Christian Lomp Locally Artinian injective hulls of simples. 3/14

8 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); 2 FBN rings (Jategaonkar); ( ) Jacobson Conjecture 3 Noetherian PI-algebras; Christian Lomp Locally Artinian injective hulls of simples. 3/14

9 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); 2 FBN rings (Jategaonkar); ( ) Jacobson Conjecture 3 Noetherian PI-algebras; 4 Noetherian semiprime rings of Krull dimension 1. Christian Lomp Locally Artinian injective hulls of simples. 3/14

10 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); 2 FBN rings (Jategaonkar); ( ) Jacobson Conjecture 3 Noetherian PI-algebras; 4 Noetherian semiprime rings of Krull dimension 1. A 1 (k) does satisfy ( ). Christian Lomp Locally Artinian injective hulls of simples. 3/14

11 Rings with ( ) Example 1 commutative Noetherian rings (Matlis); 2 FBN rings (Jategaonkar); ( ) Jacobson Conjecture 3 Noetherian PI-algebras; 4 Noetherian semiprime rings of Krull dimension 1. A 1 (k) does satisfy ( ). 5 k q [x, y] or A q 1 (k) do satisfy ( ) iff q 1 (Carvalho-Musson) Christian Lomp Locally Artinian injective hulls of simples. 3/14

12 Rings with ( ) k a field of characteristic 0. For a ring R let A 1 (R) = R[y][x; y ]. Example (Stafford) A n (k) = A 1 (A n 1 (k)) does not satisfy ( ) if [k : Q] n 1 0. Christian Lomp Locally Artinian injective hulls of simples. 4/14

13 Rings with ( ) k a field of characteristic 0. For a ring R let A 1 (R) = R[y][x; y ]. Example (Stafford) A n (k) = A 1 (A n 1 (k)) does not satisfy ( ) if [k : Q] n 1 0. Theorem (Carvalho-L.-Pusat, 2010) Let A be Noetherian algebra over k = k (+more assumptions). Then A satisfies ( ) iff A/Am does for all m Max(Z(A)). Christian Lomp Locally Artinian injective hulls of simples. 4/14

14 Rings with ( ) k a field of characteristic 0. For a ring R let A 1 (R) = R[y][x; y ]. Example (Stafford) A n (k) = A 1 (A n 1 (k)) does not satisfy ( ) if [k : Q] n 1 0. Theorem (Carvalho-L.-Pusat, 2010) Let A be Noetherian algebra over k = k (+more assumptions). Then A satisfies ( ) iff A/Am does for all m Max(Z(A)). Example (Heisenberg Lie algebras) Let h n = span(x 1,..., x n, y 1,..., y n, z) be the 2n + 1-dimensional complex Heisenberg Lie algebra with relation [x i, y i ] = z for all i. Then U(h n ) satisfies ( ) if and only if n = 1. Christian Lomp Locally Artinian injective hulls of simples. 4/14

15 Musson s example Example (Musson 82) Let g = span(x, y) with [x, y] = y. Then U(g) = k[y][x; y y ] does not satisfy ( ). Christian Lomp Locally Artinian injective hulls of simples. 5/14

16 Musson s example Example (Musson 82) Let g = span(x, y) with [x, y] = y. Then U(g) = k[y][x; y y ] does not satisfy ( ). no f.d. solvable non-nilpotent Lie algebra g over k = k, U(g) satisfies ( ) (Bohro-Gabriel-Rentschler) Christian Lomp Locally Artinian injective hulls of simples. 5/14

17 Musson s example Example (Musson 82) Let g = span(x, y) with [x, y] = y. Then U(g) = k[y][x; y y ] does not satisfy ( ). no f.d. solvable non-nilpotent Lie algebra g over k = k, U(g) satisfies ( ) (Bohro-Gabriel-Rentschler) Question (1) For which finite dimensional nilpotent g does U(g) satisfy ( )? Christian Lomp Locally Artinian injective hulls of simples. 5/14

18 Musson s example Example (Musson 82) Let g = span(x, y) with [x, y] = y. Then U(g) = k[y][x; y y ] does not satisfy ( ). no f.d. solvable non-nilpotent Lie algebra g over k = k, U(g) satisfies ( ) (Bohro-Gabriel-Rentschler) Question (1) For which finite dimensional nilpotent g does U(g) satisfy ( )? Question (2) For which derivation δ does k[y][x; δ] satisfy ( )? Christian Lomp Locally Artinian injective hulls of simples. 5/14

19 Question 1 Theorem (Hatipoğlu-L., 2012) The following are equivalent for a finite dimensional complex nilpotent Lie superalgebra g = g 0 g 1. (a) Injective hulls of simple U(g)-modules are loc. Artinian. (b) ind(g 0 ) = inf f g 0 dim(g f 0 ) dim(g 0) 2. (c) g 0 has an abelian ideal of codimension 1 or g 0 = h a where a is abelian and h is one of the following: (i) h = span(e 1, e 2, e 3, e 4, e 5 ) with [e 1, e 2 ] = e 3, [e 1, e 3 ] = e 4, [e 2, e 3 ] = e 5. (ii) h = span(e 1, e 2, e 3, e 4, e 5, e 6 ) with [e 1, e 2 ] = e 6, [e 1, e 3 ] = e 4, [e 2, e 3 ] = e 5. Christian Lomp Locally Artinian injective hulls of simples. 6/14

20 Primitive factors of superalgebras Theorem (Hatipoğlu-L., 2012) Let A be a Noetherian associative superalgebra such that 1 every primitive ideal is maximal and 2 every graded maximal ideal is generated by a normalizing sequence of generators. Then (a) injective hulls of a left simple A-module are loc. Artinian; (b) injective hulls of a left simple A/P-module are loc. Artinian for all primitive ideals P of A. Christian Lomp Locally Artinian injective hulls of simples. 7/14

21 Primitive factors of superalgebras Theorem (Hatipoğlu-L., 2012) Let A be a Noetherian associative superalgebra such that 1 every primitive ideal is maximal and 2 every graded maximal ideal is generated by a normalizing sequence of generators. Then (a) injective hulls of a left simple A-module are loc. Artinian; (b) injective hulls of a left simple A/P-module are loc. Artinian for all primitive ideals P of A. Theorem (Hatipoğlu-L., 2012) Any ideal of a finite dimensional nilpotent Lie superalgebra has a supercentralizing sequence of generators. Christian Lomp Locally Artinian injective hulls of simples. 7/14

22 Primitive factors of nilpotent Lie superalgebras Theorem (Bell-Musson 1990, Herscovich 2010) Let g be a finite dimensional nilpotent complex Lie superalgebra. 1 For f g 0 there exists a graded primitive ideal I (f ) of U(g) such that U(g)/I (f ) Cliff q (C) A p (C), where 2p = dim(g 0 /g f 0 ) and q 0. 2 For every graded primitive ideal P of U(g) there exists f g 0 such that P = I (f ). Christian Lomp Locally Artinian injective hulls of simples. 8/14

23 Primitive factors of nilpotent Lie superalgebras Theorem (Bell-Musson 1990, Herscovich 2010) Let g be a finite dimensional nilpotent complex Lie superalgebra. 1 For f g 0 there exists a graded primitive ideal I (f ) of U(g) such that U(g)/I (f ) Cliff q (C) A p (C), where 2p = dim(g 0 /g f 0 ) and q 0. 2 For every graded primitive ideal P of U(g) there exists f g 0 such that P = I (f ). We are left with classifying finite dimensional nilpotent Lie algebras g with ind(g) = dim(g) 2. Christian Lomp Locally Artinian injective hulls of simples. 8/14

24 Question 2 Example (Musson 82) k[y][x; y y ] does not satisfy ( ). Question (2) For which derivation δ does k[y][x; δ] satisfy ( )? Christian Lomp Locally Artinian injective hulls of simples. 9/14

25 ( ) for Ore extensions Theorem (Carvalho-Hatipoğlu-L., 2012) Let R be a commutative Noetherian domain over k. Set S = R[x; δ] for some δ Der k (R). Suppose that 1 R is not δ-simple; 2 R is δ-primitive (D.Jordan), i.e. a maximal ideal m of R that does not contain any non-zero δ-ideal. 3 every non-zero δ-ideal contains a non-zero Darboux element, i.e. an element a with Ra being an δ-ideal. Christian Lomp Locally Artinian injective hulls of simples. 10/14

26 ( ) for Ore extensions Theorem (Carvalho-Hatipoğlu-L., 2012) Let R be a commutative Noetherian domain over k. Set S = R[x; δ] for some δ Der k (R). Suppose that 1 R is not δ-simple; 2 R is δ-primitive (D.Jordan), i.e. a maximal ideal m of R that does not contain any non-zero δ-ideal. 3 every non-zero δ-ideal contains a non-zero Darboux element, i.e. an element a with Ra being an δ-ideal. Then 0 S/Sm S/Sm(x 1) S/S(x 1) 0 is a non-artinian essential extension of the simple module S/Sm, i.e S does not satisfies ( ). Christian Lomp Locally Artinian injective hulls of simples. 10/14

27 k[y][x; α, δ] Corollary k[y][x; δ] satisfies ( ) iff δ = λ y Corollary (Carvalho-Hatipoğlu-L., 2012) for some λ k. The following are equivalent for an automorphism α and an α-derivation δ of k[y]: 1 injective hulls of simple k[y][x; α, δ]-modules are locally Artinian; 2 α id has finite order or α = id and δ is locally nilpotent. 3 k[y][x; α, δ] is isomorphic to A q 1 (k) or k q[x, y] for q 1. Christian Lomp Locally Artinian injective hulls of simples. 11/14

28 Locally nilpotent derivations Theorem (Carvalho-Hatipoğlu-L., 2012) Injective hulls of simple R[x; δ]-modules are locally Artinian provided δ is locally nilpotent and R is an affine commutative k-algebra. Christian Lomp Locally Artinian injective hulls of simples. 12/14

29 Locally nilpotent derivations Theorem (Carvalho-Hatipoğlu-L., 2012) Injective hulls of simple R[x; δ]-modules are locally Artinian provided δ is locally nilpotent and R is an affine commutative k-algebra. Sketch: Set a = span({δ i (y j ) i 0, 1 j n}) and g = kx a R[x; δ]. Set [x, δ i (y j )] = δ i+1 (y j ) i, j Then U(g) R[x; δ] and as g is nilpotent and a is an abelian ideal of codimensiom 1, U(g) and hence R[x; δ] satisfies ( ). Christian Lomp Locally Artinian injective hulls of simples. 12/14

30 Locally nilpotent derivations II Lemma Let R be a domain with locally nilpotent derivation δ and y R with δ(y) = 1. Set S = R[x; δ]. For every a R \ R δ consider 0 S/S(x + a) S/S(x + a)x S/Sx 0. Then S/S(x + a) embeds essentially in S/S(x + a)x and Kdim( S S/Sx) = Kdim( R δr δ ). Christian Lomp Locally Artinian injective hulls of simples. 13/14

31 Locally nilpotent derivations II Lemma Let R be a domain with locally nilpotent derivation δ and y R with δ(y) = 1. Set S = R[x; δ]. For every a R \ R δ consider 0 S/S(x + a) S/S(x + a)x S/Sx 0. Then S/S(x + a) embeds essentially in S/S(x + a)x and Kdim( S S/Sx) = Kdim( R δr δ ). Hence if S(x a) is maximal in S and R δ is not a division ring, then S does not satisfiy ( ). Christian Lomp Locally Artinian injective hulls of simples. 13/14

32 Locally nilpotent derivations II Lemma Let R be a domain with locally nilpotent derivation δ and y R with δ(y) = 1. Set S = R[x; δ]. For every a R \ R δ consider 0 S/S(x + a) S/S(x + a)x S/Sx 0. Then S/S(x + a) embeds essentially in S/S(x + a)x and Kdim( S S/Sx) = Kdim( R δr δ ). Hence if S(x a) is maximal in S and R δ is not a division ring, then S does not satisfiy ( ). Example (Stafford, cf. Coutinho, Bernstein-Lunts) For D = A n 1 (k) there exists a D[y] such that x + a generates a maximal left ideal in A n (k) = A 1 (D) = D[y][x; y ]. R = A 1 (Q)[y] satisfies ( ) while A 2 (Q) = R[x; y ] doesn t. Christian Lomp Locally Artinian injective hulls of simples. 13/14

33 Thank you Christian Lomp Locally Artinian injective hulls of simples. 14/14

MONOLITHIC MODULES OVER NOETHERIAN RINGS

MONOLITHIC MODULES OVER NOETHERIAN RINGS PAULA A.A.B. CARVALHO IAN M. MUSSON Abstract. We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl