Associative rings

The local finite basis property and local representability of varieties of associative rings

“…b) Moreover, composition of pushing replacements can be performed arbitrarily many times. 6 Hence there is a well-defined non-zero infinite composite P ∞ (f ) of pushing replacements from T(f ) (the corresponding construction is described below ).…”

Локальная Конечная Базируемость И Локальная Представимость Многообразий Ассоциативных Колец

“…Тогда существует ненулевой элемент P 2 (f ) из T(f ), имеющий вид, указанный в лемме 4.9 (или предложении 4.20). б) Более того, композицию расталкивающих замен можно проводить любое число раз 6 , так что определена ненулевая бесконечная композиция P ∞ (f ) расталкивающих замен из T(f ) (соответствующая конструкция определена ниже).…”

Monomial algebras

“…We put in lexicographic order the vectors (M1,..., Mk) and demonstrate that elements which are linearly nonrepresentable by elements with smaller vectors have the required form. Let us choose in (2) (3) holds, then there exists a constant C such that S(q) < Cq k-x.…”

“…We associate the empty word with the empty word, words baib E Wd (a, b) with generators hi, and consider the algebra A ^ = f(a,b)/I, where I is the ideal generated by the following elements: (1) b 3, (2) aba, (3) words which correspond to the zero word in the algebra A under the substitution baib ~ hi.…”

“…(1) F(A) does not have any twice cyclic vertices; (2) A has a polynomial growth; (3) A has a nonezponential growth; (4) A can be represented by matrices over a field; (5) a polynomial identity holds in A.…”