Leonid A. Bokut, Victor Latyshev, Ivan Shestakov, Efim's Selected works of A.I. Shirshov PDF

By Leonid A. Bokut, Victor Latyshev, Ivan Shestakov, Efim Zelmanov, Murray Bremner, Mikhail V. Kotchetov

ISBN-10: 3764388579

ISBN-13: 9783764388577

This booklet provides translations of chosen works of the well-known Russian mathematician A.I. Shirshov (1921–1981). He was once a pioneer in different instructions of associative, Lie, Jordan, and replacement algebras, in addition to teams and projective planes. His identify is linked to notions and effects on Gröbner-Shirshov bases, the Composition-Diamond Lemma, the Shirshov-Witt Theorem, the Lazard-Shirshov removal procedure, Shirshov’s peak Theorem, Lyndon-Shirshov phrases, Hall-Shirshov bases, Shirshov’s theorem at the Kurosh challenge for substitute and Jordan algebras, and Shirshov’s theorem at the speciality of Jordan algebras with turbines. Shirshov’s rules have been utilized by his scholar Efim Zelmanov for the answer of the limited Burnside challenge. numerous recognized algebraists supply during this ebook specific reviews at the impression of Shirshov’s paintings on present algebra.

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Example text

2. Step 2. Now suppose that the lemma holds for some pair of words α1 = at Dar , β1 = ak bs . We will show that in this case the lemma also holds for the words α2 = at Dak , β2 = ar bs . Indeed, α∗2 ◦ β2∗ = (at Dak )∗ ◦ (ar ◦ bs ) = 2[(at D)∗ ◦ ak ] ◦ (ar ◦ bs ) − (at+k D)∗ ◦ (ar ◦ bs ) = −2J1 {(at D)∗ , ak , ar , bs } + 2[(at D)∗ ◦ ak+r ] ◦ bs + 2[(at D)∗ ◦ (ak ◦ as )] ◦ ar + 2[(at D)∗ ◦ (ar ◦ bs )] ◦ ak − 2[(at D)∗ ◦ bs ] ◦ ak+r − 2[(at D)∗ ◦ ar ] ◦ (ak ◦ bs ) − (at+k D)∗ ◦ (ar ◦ bs ) = − 2J1 {at D, ak , ar , bs } + 2(at D ◦ ak+r ) ◦ bs + 2[at D ◦ (ak ◦ bs )] ◦ ar + 2[at D ◦ (ar ◦ bs )] ◦ ak − 2(at D ◦ bs ) ◦ ak+r − (at+r D) ◦ (ak ◦ bs ) − (at+k D) ◦ (ar ◦ bs ) ∗ − (at Dar )∗ ◦ (ak ◦ bs ) = −α∗1 ◦ β1∗ + 2(at D ◦ ar ) ◦ (ak ◦ bs ) + 2(at D ◦ ak ) ◦ (ar ◦ bs ) − (at+r D) ◦ (ak ◦ bs ) − (at+k D) ◦ (ar ◦ bs ) ∗ = −α∗1 ◦ β1∗ + at Dar ◦ (ak ◦ bs ) + at Dak ◦ (ar ◦ bs ) = −α∗1 ◦ β1∗ + (α1 ◦ β1 )∗ + (α2 ◦ β2 )∗ = (α2 ◦ β2 )∗ .

Lemma 6. Any associative ring Σ with characteristic different from 2 can be embedded in a ring Σ that admits unique division by 2. Proof. Consider the set Σ of pairs (σ, 2k ) where σ ∈ Σ, and k ≥ 0 is an integer. We will consider the pairs (σ1 , 2k1 ) and (σ2 , 2k2 ) to be equivalent if 2k2 σ1 = 2k1 σ2 . We define addition and multiplication of the pairs in the familiar way: (σ1 , 2k1 ) + (σ2 , 2k2 ) = 2k2 σ1 + 2k1 σ2 , 2k1 +k2 , (σ1 , 2k1 )(σ2 , 2k2 ) = σ1 σ2 , 2k1 +k2 . Obviously, the ring Σ satisfies the requirements of Lemma 6.

In the case of algebras over a field, Cohn [2] proved that a homomorphic image of a special Jordan algebra is not necessarily special. It follows that the class of special Jordan algebras cannot be defined by identical relations. In the present paper, it is proved that a Jordan algebra over Σ that has a finite or countably infinite set of generators is special if and only if it can be embedded into a Jordan algebra over Σ with two generators. In the last section of this paper, we remove the requirement that for every element a there exists an element b such that 2b = a.

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Selected works of A.I. Shirshov by Leonid A. Bokut, Victor Latyshev, Ivan Shestakov, Efim Zelmanov, Murray Bremner, Mikhail V. Kotchetov


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