We define a notion of associative representation for algebras. We prove the existence of faithful associative representations for any alternative, Mal ’cev, and Poisson algebra, and prove analogs of Ado-Iwasawa theorem for each of these cases. We construct also an explicit associative representation of the Cayley–Dickson algebra in the matrix algebra [Formula: see text]
Abstract. We prove that the prime radical rad M of the free Malcev algebra M of rank more than two over a field of characteristic = 2 coincides with the set of all universally Engelian elements of M. Moreover, let T (M) be the ideal of M consisting of all stable identities of the split simple 7-dimensional Malcev algebra M over F . It is proved that rad M = J(M) ∩ T (M), where J(M) is the Jacobian ideal of M. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras. An algebra M is called a Malcev algebra if it satisfies the identitieswhere J(x, y, z) = (xy)z + (zx)y + (yz)x is the Jacobian of the elements x, y, z [7,9,5]. Since for a Lie algebra the Jacobian of any three elements vanishes, Lie algebras fall into the variety of Malcev algebras. Among the non-Lie Malcev algebras, the traceless elements of the octonion algebra with the product given by the commutator [x, y] = xy − yx is one of the most important examples [9,5,6]. In 1977 I. P. Shestakov [11] proved that the free Malcev algebra M n on n ≥ 9 free generators is not semiprime; that is, M n contains nonzero nilpotent ideals. In 1979, V. T. Filippov [3] extended this result to free Malcev algebras with more than four generators. Therefore, the prime radical rad M n = 0 for n > 4, and a natural question on the description of this radical arises.For free alternative algebras, it was proved by Shestakov in [10] that the prime radical coincides with the set of nilpotent elements. A similar fact was established by E. Zel manov [15] for free Jordan algebras.In anticommutative algebras, the role of nilpotent elements is played by engelian elements. An element a of an algebra M is called engelian if the operator of right multiplication R a : x → xa is nilpotent. We will call an element a ∈ M universally engelian if, for every algebra M ⊇ M , the element a is engelian in M . In other words, the image R a of the element a in the (associative) universal multiplicative enveloping algebra R(M ) of M is nilpotent. In the present paper, we prove that the
We find a basis of the free Malcev algebra on three free generators over a field of characteristic zero. The specialty and semiprimity of this algebra are proved. In addition, we prove the decomposability of this algebra into subdirect sum of the free Lie algebra rank three and the free algebra of rank three of variety of Malcev algebras generated by a simple seven-dimensional Malcev algebra
We continue studying the associative rings complete and reduced in the sense of Martynov. We prove that the group of invertible elements of a reduced associative ring is reduced. We compute the complete radical of a group ring over the ring of integers and the complete radical of the group algebra over an arbitrary algebra over a finite prime field.The notions of completeness (divisibility) and reducibility play an important role in the theory of abelian groups. It turns out that there is another approach to these notions that uses the theory of varieties of groups. This fact enabled Martynov to define some analogs of these notions in the general universalalgebraic situation [1]. The problems here are of interest for universal as well as "classical" algebras (groups, semigroups, modules, rings, etc.). The notions of completeness and reducibility were studied for modules in [2][3][4]; for linear algebras, in [5]; for semigroups, in [6,7]; and for associative rings, in [8][9][10].In line with [11], the definition of complete associative ring may be given as follows: Let M be a set of varieties. An associative ring is called M -complete, if it has no nonzero homomorphisms to the rings in M . In particular, if M consists of the atoms of the lattice of varieties of rings then an M -complete ring is called complete. Recall that these varieties are exhausted by the following series: the varieties Z p given by the identities xy = 0 and px = 0 and the varieties F p with the identities px = 0 and x p − x = 0 over all prime p. An associative ring is called reduced if it contains no nonzero complete subrings. From here on, by a ring we mean an associative ring.As Martynov noted in [11], each ring R possesses the inclusion greatest complete subring C(R) which is an ideal and the map R → C(R) in the abstract class of rings which is a strong radical in the sense of Kurosh [12]. This radical is called complete. It is characterized in [9] for group algebras over finite prime fields.The atoms of the lattice of varieties of groups are well known to be exhausted by the series of varieties A p of abelian groups of exponent p over all prime p. Following [13], we say that a group G is A p -complete if it has no homomorphisms to the nonidentity groups in A p . It is easy to see that G is A p -complete if and only if G G p = G. Let π be a set of primes. A group is called π-complete if it is A p -complete for each prime p from π. Using Remark 3 from [13], it is easy to show that, for every set π of primes, each group contains the greatest π-complete subgroup which is a normal divisor. If π coincides with the set of all primes then the group is called complete. A group is called reduced if it has no nontrivial complete subgroups. The greatest complete subgroup of G is denoted by C(G).The main results of this paper are as follows:Theorem 1. If a unital ring R is reduced then so is the group R * .Theorem 2. Let G be an arbitrary group. Then C(ZG) = w(C(G)).Theorem 3. Let R be an arbitrary algebra over GF (p) and let G be an arbitrary group. ...
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