Counterexamples to the Specht problem

Belov, A. Ya.

doi:10.1070/sm2000v191n03abeh000460

Cited by 39 publications

(18 citation statements)

References 12 publications

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“…Otherwise, f is called wild. Consequently, in view of the cyclic property of φ, a double φ-word ϕ n (1,2,3,4,5,6) satisfies the assertion of Lemma 5.3 for the variables…”

Section: Tame Wordsmentioning

confidence: 91%

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Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Kuzmin¹

2015

Communications in Algebra

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Since 1976, it is known from the paper by V. P. Belkin that the variety RA 2 of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains non-finitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA 2 -algebra of finite rank r over a field F , for char(F) = 2, is Spechtian iff r = 1. We construct a non-finitely based variety M generated by the Grassmann V-algebra of rank 2 of certain finitely based subvariety V ⊂ RA 2 over a field F , for char(F) = 2, 3, such that M can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.Key words: non-finitely based variety of algebras, Spechtian variety of algebras, right alternative metabelian algebra, superalgebra, Grassmann algebra.

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“…Otherwise, f is called wild. Consequently, in view of the cyclic property of φ, a double φ-word ϕ n (1,2,3,4,5,6) satisfies the assertion of Lemma 5.3 for the variables…”

Section: Tame Wordsmentioning

confidence: 91%

“…Ya. Belov [2], A. V. Grishin [4], and V. V. Schigolev [18] constructed, independently, non-finitely based varieties of associative algebras over a field of prime characteristic.…”

Section: Introductionmentioning

confidence: 99%

Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Kuzmin¹

2015

Communications in Algebra

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“…(3) Let a ∈ M be such that f (a) = 0. Then f k (a + z k M) = 0 + z k N, so a ∈ z k M, but then (2) implies that a = 0.…”

Section: Conclusion Of the Solution Of Specht's Problem For Arbitary mentioning

confidence: 99%

Specht’s problem for associative affine algebras over commutative Noetherian rings

Belov-Kanel¹,

Rowen²,

Vishne³

2015

Trans. Amer. Math. Soc.

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In a series of papers [6], [7], [8] we introduced full quivers and pseudoquivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "q-characteristic coefficient-absorbing polynomial in each T-ideal Γ," i.e., a non-identity of the representable algebra A arising from Γ, whose ideal of evaluations in A is closed under multiplication byq-powers of the characteristic coefficients of matrices corresponding to the generators of A, whereq is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring C involves localizing at finitely many elements a kind of C, and reducing to the field case by a local-global principle.

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“…Nevertheless this result is no longer true in the case of positive characteristic even when the ground field is infinite. For instance see the counter-examples given by Belov [4] in any characteristic, by Grishin [21] in characteristic 2 and by Shchigolev [31]. It is rather difficult to provide a finite basis of T (A) for a given algebra A. Genov [18,17] and Latyshev [27] proved that every algebra satisfying the identities of UT n (F ) has a finite basis of its polynomial identities.…”

Section: Introductionmentioning

confidence: 98%

“…where the variables y, y 1 , y 2 , y 3 and y 4 have Z 2 -degree 0 and the variables z, z 1 , z 2 , z 3 and z 4 have Z 2 -degree 1.…”

Section: Introductionmentioning

confidence: 99%

On Z2-graded identities of UT2(

Centrone

Silva

2015

Linear Algebra and its Applications

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Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F . We consider the upper triangular matrix algebra UT 2 (E) with entries in E endowed with the Z 2 -grading inherited by the natural Z 2 -grading of E and we study its ideal of Z 2 -graded polynomial identities (T Z2 -ideal) and its relatively free algebra. In particular we show that the set of Z 2 -graded polynomial identities of UT 2 (E) does not depend on the characteristic of the field. Moreover we compute the Z 2 -graded Hilbert series of UT 2 (E) and its Z 2 -graded Gelfand-Kirillov dimension.

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Counterexamples to the Specht problem

Cited by 39 publications

References 12 publications

Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Specht’s problem for associative affine algebras over commutative Noetherian rings

On Z2-graded identities of UT2(

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