Lie structure in semi - prime rings with involution

Lanski, Charles

doi:10.1080/00927877608822134

Cited by 21 publications

(18 citation statements)

References 7 publications

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“…Consequently, if fe G i£, then (a£a)F(a£a) = 0 and we have ak G i^ with (aka) 2 = 0. It follows [5,Lemma 7] that aka = 0. Therefore a(r -r*)a = 0 for any r £ R, or equivalently, ara = ar*a.…”

Section: Lemma 1 Let a Be A Subring Of R Contained In S And Satisfyimentioning

confidence: 99%

“…The next result is implicit in [5] and is used to obtain another technical lemma required in Theorem 1. Recall that * is said to be of the first kind if Z C S, and of the second kind otherwise.…”

Section: Lemma 1 Let a Be A Subring Of R Contained In S And Satisfyimentioning

confidence: 99%

“…Proof. Applying [5,Lemma 3 and Sublemma], either V contain a nonzero ideal of 7^ or x 2 G Z for all x G V. If * is of the second kind then [5,Lemma 6] implies that R satisfies Si or K C Z, which in turn implies that R is commutative, since 2tR C K + tK for any / G Z H K. Finally, if * is of the first kind, then the proof of [5,Lemma 9] shows that R satisfies 5 4 or else one may reduce to the case where R is simple and Z^0. Now an examination of [1,[36][37][38][39][40] shows that when R is simple V must be 3-dimensional over Z and have a nonzero centralizer in K, disjoint from V and contained in V\ This absurdity shows that R must satisfy 5 4 .…”

Section: Lemma 1 Let a Be A Subring Of R Contained In S And Satisfyimentioning

confidence: 99%

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Invariant Subrings in Rings with Involution

Lanski

1978

Can. j. math.

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The purpose of this paper is to consider, in rings with involution, the structure of those subrings which are invariant under Lie commutation with [K, K]. Our goal is to find conditions which force such subrings to contain a noncentral ideal of the ring. Of course, the subring itself may lie in the center. Orders in 4 × 4 matrix rings over fields are known to provide examples of invariant subrings which are not central and contain no ideal (see [1, 40] or [5]). Except for subdirect products of these two kinds of “counter-examples”, we show that in semi-prime rings, invariant subrings do contain noncentral ideals. This generalizes work of Herstein [4] in two directions by considering semi-prime rings rather than simple rings, and by using [K, K] instead of K.

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Section: Lemma 1 Let a Be A Subring Of R Contained In S And Satisfyimentioning

confidence: 99%

Section: Lemma 1 Let a Be A Subring Of R Contained In S And Satisfyimentioning

confidence: 99%

Section: Lemma 1 Let a Be A Subring Of R Contained In S And Satisfyimentioning

confidence: 99%

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Invariant Subrings in Rings with Involution

Lanski

1978

Can. j. math.

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“…Baxter [1] first studied in associative algebras with involution the Lie structure of K and [K, K], and after several authors contributed to this subject: T.E. Erickson [2], C. Lanski [13], W.S. Martindale III and C.R.…”

Section: Introductionmentioning

confidence: 99%

The derived superalgebra of skew elements of a semiprime superalgebra with superinvolution

Laliena

2014

Journal of Algebra

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Available online xxxx Communicated by Louis Rowen MSC: 16W55 17A70 17C70 Keywords: Associative superalgebras Semiprime superalgebras Superinvolutions Skewsymmetric elements Lie structureIn this paper we investigate the Lie structure of the derived Lie superalgebra [K, K], with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of [K, K], then either there exists an ideal J of A such that the Lie ideal [J ∩ K, K] is nonzero and contained in U , or A is a subdirect sum of A , A , where the image of U in A is central, and A is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.

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“…Baxter (see [1]). Afterwards, several authors have made different contributions and generalizations to the subject (see for instance [2,9,11]). …”

Section: Introductionmentioning

confidence: 99%

Lie structure in semiprime superalgebras with superinvolution

Laliena

Sacristán

2007

Journal of Algebra

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In this paper we investigate the Lie structure of the Lie superalgebra K of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of K, then either there exists an ideal J of A such that the Lie ideal [J ∩ K, K] is nonzero and contained in U , or A is a subdirect sum of A , A , where the image of U in A is central, and A is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.

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Lie structure in semi - prime rings with involution

Cited by 21 publications

References 7 publications

Invariant Subrings in Rings with Involution

Invariant Subrings in Rings with Involution

The derived superalgebra of skew elements of a semiprime superalgebra with superinvolution

Lie structure in semiprime superalgebras with superinvolution

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