Nondegenerate Jordan Algebras Satisfying Local Goldie Conditions*

Transcription

1 Ž. OURNAL OF ALGEBRA 182, ARTICLE NO Nondegenerate ordan Algebras Satisfying Local Goldie Conditions* Antonio Fernandez Lopez and Eulalia Garcıa Rus Departamento de Algebra, Geometrıa y Topologıa, Facultad de Ciencias, Uniersidad de Malaga, Malaga, Spain Communicated by Walter Feit Received May 20, 1994 A structure theorem is given for nondegenerate ordan algebras satisfying the ascending chain condition on annihilators of a single element and such that contains no infinite direct sum of inner deals inside the inner ideal generated by each element. As a consequence of this theorem and of the main results of a previous paper by the authors Ž. Algebra 174 Ž 1995., , it is obtained that such ordan algebras are precisely the local orders in nondegenerate ordan algebras satisfying dcc on principal inner ideals and without non-artinian quadratic ideals, which etends to local orders the Zel manov theorem for Goldie ordan algebras Academic Press, Inc. 1. PRELIMINARIES We remind the reader of the basic quadratic notations; as references we mention the books 4, 5. A Ž quadratic. ordan algebra Ž, U, 2. over an arbitrary ring of scalars is a -module with quadratic maps U : End Ž. and 2 : Ž squaring. satisfying the following identities in all scalars etensions: V, y 2 y Ž 1. U Ž y. Uy Ž U Ž. Ž y Ž. Ž. UU Uy 4 * This work is supported by DGICYT Grant PC and by the Plan Andaluz de Investigacion y Desazzollo Tecnologico $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 52

2 NONDEGENERATE ORDAN ALGEBRAS 53 U 2 U 2 Ž 5. U UUU. Ž 6. Uy y Here y Ž y. 2 2 y 2 and V zu yyz4,y,z Uzy 1 UyUy. In the case that, Ž quadratic. z 2 ordan algebras can be characterized aiomatically as the linear ordan algebras with product y 1 Ž y. satisfying 2 y y, Ž 2 y. 2 Ž y.. Recall that an inner ideal of is a submodule M of such that UM M.If then U is an inner ideal, the principal inner ideal determined by. Following 5 or 8, the annihilator of any subset X of a ordan algebra is the subset ann Ž X. annž X. of all elements z satisfying UUzV z z,v,zuu z UU z0 for all X. Properties of annihilators are listed in 8. In particular, annž X. is an inner ideal of for each subset X of. For a linear ordan algebra, ann Ž X. coincides with that defined by Zel manov, i.e., 4 ann Ž X. z : z X Ž z,, X. 0, where Ž a, b, c. Ž a b. c a Ž b c. is the associator of a, b, c. A ordan algebra is nondegenerate if Ua 0 implies a 0. When is nondegenerate the annihilator of an ideal I of has an easy epression Žsee. 7 annž I. a : UI04 Ž 7. and satisfies a I annž I. 0. Ž 8. An easy consequence of the above results is the following: LEMMA 1. Let be a nondegenerate ordan algebra, B an ideal of, and annž B. the quotient algebra with canonical projection : of onto. Then Ž. i is nondegenerate, Ž ii. for B and z, z ann Ž. if and only if z ann Ž.. As a consequence of Ž ii., if satisfies acc on ann Ž., B, then satisfies acc on ann Ž., B. Proof. Ž. i Let be such that U 0. Then UBBannŽ B. 0 Ž. Ž. Ž. by 8, which implies ann B by 7.

3 54 FERNANDEZ LOPEZ AND GARCIA RUS Ž ii. Clearly, if z ann Ž. then z ann Ž.. Conversely, if z ann Ž. then z annihilates module annž B., but B. Hence z annihilates module annž B. B 0byŽ 8.. A ordan algebra is prime if UC0 B implies B 0orC0, for B, Cideals of. Following 7 a nondegenerate ordan algebra is prime if and only if it has orthogonality intersection, that is, for any non-zero ideals B, C of its intersection B C is non-zero. Moreover, for nondegenerate ordan algebras, primeness is inherited by ideals. 2. UNIFORM IDEALS Let be a ordan algebra. An ideal I of will be called uniform if for any non-zero ideals B, C of inside I, the intersection B C is non-zero. THEOREM 1. Let be a nondegenerate ordan algebra. Then Ž. i A non-zero ideal I of is uniform if and only if its annihilator annž I. is maimal among all annihilators annž B. with B being a non-zero ideal of. Ž ii. For each uniform ideal I of there eists a maimal uniform ideal M of containing I, actually M annžannž I... Ž iii. The sum of all maimal uniform ideals of is direct. Proof. Ž. i Suppose first that I is a uniform ideal. We note that for every non-zero ideal B of inside I, annž B. I 0. Otherwise, annž B. I 0 would imply that B annž B. B ŽI annž B.. 0 by uniformity of I, which contradicts Ž. 8. Now annž B. I 0 implies that annž B. annž I.. Suppose that C is a non-zero ideal of such that annž I. annž C.. If CI0 then C annž I. annž C. and hence C 0 by nondegeneracy of, which is a contradiction, so C I is a non-zero ideal contained in I. Then we have by the first part of the proof that annž C. annž C I. annž I., and hence annž C. annž I., as required. Suppose now that annž I. is maimal among all the annihilators ideals annž D., D a non-zero ideal of. Let B and C be non-zero ideals of inside I. Since B I, annž I. annž B. and hence annž I. annž B. by maimality of annž I.. Now B C 0 would imply that C I annž I. 0, which is a contradiction. Ž ii. Note that annžannž I.. is uniform because annžannžannž I... annž I. which is maimal by Ž i.. Let B be a uniform ideal of containing I.

4 NONDEGENERATE ORDAN ALGEBRAS 55 Then annž B. annž I. implies annž B. annž I. by maimality of annž B. and hence B annžannž B.. annžannž I... Ž iii. Let M M 4 be the family of all maimal uniform ideals of. Take M M. Then ž / Ý Ý Ý M M M M M M 0 since if M M were non-zero then annž M. annž M M. would imply that Ž. Ž. ann M ann M M annž M. by maimality, and hence M annžannž M.. annžannž M.. M, which is a contradiction. Now M Ž Ý M. 0 implies that M Ž Ý M. 0 since is nondegenerate. COROLLARY 1. Let be a nondegenerate ordan algebra and I a uniform ideal of. Then the quotient algebra annž I. is a prime nondegenerate ordan algebra. Proof. Let B, C be ideals of such that B C ann Ž I.. If both B and C are not contained in ann Ž I., then B I and C I are non-zero, and hence B C I 0 by uniformity of I, which is a contradiction because I ann Ž I. 0. Thus one of these ideals B or C is contained in ann Ž I., as required. 3. UNIFORM ELEMENTS Since annihilators are inner ideals, for any element u in a ordan algebra, annž u. annž. for U. u Now a non-zero element u is called uniform if annž u. annž. for any 0 U. u Notice that if u is uniform then every non-zero element in the inner ideal generated by u is uniform as well. Clearly every non-zero element u such that annž u. is maimal is a uniform element. In particular, if satisfies acc on the annihilators of its elements, then every non-zero inner ideal of contains a uniform element. Moreover, by using socle theory Žsee. 6 it is not difficult to see that an element in the socle of is uniform if and only if is minimal, that is, it generates a minimal inner ideal.

5 56 FERNANDEZ LOPEZ AND GARCIA RUS PROPOSITION 1. Let be a nondegenerate ordan algebra. Then eery uniform element u generates a uniform ideal. Proof. Let I Iu Ž. denote the ideal of generated by u and let B, C be non-zero ideals of contained in I. We must prove that B C is non-zero. Note first that UB u and similarly UC u are both non-zero. Otherwise, UB0would imply u annž B., and hence B I annž B. u, which is a contradiction. Thus there eists b B such that Ub0, u so annž u. annž Ub. by uniformity of u. Hence, if B C were zero, then u C annž B. annž Ub. annž u. which would imply that u annž C., which is a contradiction as shown above. Let be nondegenerate and u uniform. The maimal uniform ideal Mu Ž. of containing u will be called the uniform component of u, and the sum Ý Mu Ž. of all the uniform components will be called the foundation of and denoted by FŽ.. Notice that by Ž iii. of Theorem 1, the sum Ý Mu Ž. is direct. Remark 1. There eist further elements generating uniform ideals. In fact, it can be shown that every element of a nondegenerate ordan algebra generating a uniform inner ideal, that is, two non-zero inner ideals of inside U have always non-zero intersection, also generates a uniform ideal. However, uniform elements work better than those elements generating uniform inner ideals. Following 3, an essential subdirect product of a collection 4 of ordan algebras is any subdirect product of the which contains an essential ideal of the direct product of the.if is actually contained in the direct sum of the, then will be called an essential subdirect sum. u 4. NONDEGENERATE ORDAN ALGEBRAS WITH LOCAL GOLDIE CONDITIONS Given a subset X of a ordan algebra, write K Ž X. to denote the inner ideal of generated by X. A direct sum of inner ideals I 4 is defined by the condition that I K Ž Ý I. 0 for each I. An element is said to have finite Goldie dimension if does not contain infinite direct sums of inner ideals inside U. LEMMA 2. Let B be an ideal of a nondegenerate ordan algebra. If a has finite Goldie dimension then a has also finite Goldie dimension in annž B..

6 NONDEGENERATE ORDAN ALGEBRAS 57 Proof. Let Ý I be a direct sum of inner ideals of inside U. For 1 each set I Ž I. which is an inner ideal of. We claim that the inner ideal L I UBis non-zero. Indeed, take a non-zero element a I. Since I U, there eists y such that Ž a. Ž Uy.. Hence Uy is an element of I U which is not in annž B.. Then L I UBis a non-zero inner ideal. We are going to show that the sum of inner ideals Ý L is direct. Indeed, for each inde, ž Ý / ž Ý / L K L I K I B Ž. Ž. but I K Ý I ann B because the sum Ý I is direct. Hence ž Ý / L K L BannŽ B. 0. Since has finite Goldie dimension, this direct sum of inner ideals must be finite, which proves that has also finite Goldie dimension, as required. THEOREM 2. Let be a nondegenerate ordan algebra such that eery non-zero ideal of contains a uniform element. Then is an essential subdirect product of prime nondegenerate ordan algebras each of which contains a uniform element. In fact, the foundation of is an essential ideal of the direct product of the. If additionally eery element has finite Goldie dimension, then is actually an essential subdirect sum of the and each element has finite Goldie dimension. Proof. Let FŽ. be the foundation of. Then annžfž.. 0 since otherwise it would contain a uniform element u, and hence by Proposition 1, Iuann Ž. ŽFŽ.. annž Iu Ž.. which is a contradiction. Now annžfž.. Mu0 Ž. implies that is a subdirect product of the ordan algebras annž M. where M ranges over the family of all uniform compo- nents of. Clearly M is an essential ideal of Ł. Since by Corollary 1 each is prime and nondegenerate, we must now show that contains a uniform element, but this is clear since if u M is uniform in then u is uniform in by Lemma 1Ž ii.. Suppose now that every element has finite Goldie dimension. We claim that annž M. up to at most for a finite set of indices. Suppose on the contrary that there eists an infinite sequence 4 of indices such n

7 58 FERNANDEZ LOPEZ AND GARCIA RUS that annž M. for all positive integers n. Then n 0 I : UM n n form an infinite direct sum of inner ideals of inside U, which contra- dicts that has finite Goldie dimension. This proves the claim. Now it follows from Lemma 2 that every element has finite Goldie dimension, as required. Remark 2. A statement similar to Theorem 2 was proved in 2, for the case of an associative algebra, by using powerful techniques of ring theory. We stress that our approach is elementary and allows us also to get the associative result. 5. LOCAL GOLDIE THEOREM FOR ORDAN ALGEBRAS For definitions not eplicitly stated in this section the reader is referred to 1. THEOREM 3. A linear ordan algebra is a local order in a nondegenerate ordan algebra Q satisfying dcc on principal inner ideals and such that Q does not contain non-artinian quadratic ideals, if and only if is nondegenerate, with acc on annihilators of a single element, and such that each of its element has finite Goldie dimension. Proof. Suppose that is a nondegenerate ordan algebra satisfying acc on ann Ž. and such that each of its elements has finite Goldie dimension, i.e., for each a, contains no infinite direct sum of inner ideals inside U. a By the acc on annihilators, each non-zero inner ideal I of contains a uniform element, so, by Theorem 2, M Ž 9. with M 4 being the family of all uniform components of, and M, where Žisomorphic to ann Ž M.. is a prime nondegenerate ordan algebra containing uniform elements and such that each of its elements has finite Goldie dimension. Then by 1, Theorem 25, each is a local order in a simple ordan algebra Q containing minimal inner ideal and which is not a non-artinian quadratic factor. Hence, by 1, Theorem 23, M is also a local order in Q. Then it is easy to see that M is a local order in the ordan algebra Q Q, which is nondegenerate and agrees with its socle, equivalently satisfies dcc on principal inner ideals. Now Ž. 9 implies, by 1, Ž 4.1., that is a weak local order, and hence a local order by 1, Theorem 20 in Q. The converse follows from 1, Theorem 20.

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