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Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings.Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings. I. Infinite-DimensionalLie Algebras I. Books. In the Russian language there appeared during this period the successive parts of the book Lie Algebras of the series of Bourbaki [26, 27], the book of Dixmier [55], and the book of Kaplansky [91]. Two books are devoted to the theory of infinite-dimensional Lie algebras: Amayo and Stewart [240] and Bakhturin [257]. The books [365, 400, 716, 718] are written in a more traditional style. Universal enveloping algebras are also discussed in [284].2. Varieties of Lie Algebras. Although the first works properly devoted to varieties of Lie algebras appeared only in 1967 (V. A. Parfenov) and in 1968 (Yu. A. Bakhturin), the machinery and first major results of this theory actually appeared earlier in the works of A. I. Mal'tsev, A. L Kostrikin, and A. I. Shirshov. Moreover, in the hands of these specialists in this theory the theories of varieties of groups and associative algebras were advanced. The latter circumstance led to the situation that at first analogues of results in these theories were sought in varieties of Lie algebras. Thus, theorems were proved on the freeness of a semigroup of varieties of Lie algebras over a field of characteristic 0 [148], on the description of Sehreier varieties over such a field [ii], and a monotonicity theorem for verbal ideals [ii, 146, 49]. It was noted in [14] that the theorems on a semigroup of varieties and Sehreier varieties are also true in the case of an arbitrary infinitedimensional field; it is also shown there that in the case of a field of two elements the multiplication of varieties of Lie algebras is nonassociative. A description of Schreier varieties of Lie algebras, i.e., varieties in which any subalgebra of a free algebra is free, was given by M. V. Zaitsev for the case of any commutative ring: it is shown there that the Sehreier varieties are trivial; e.g., if the base ring is a field these are the varieties D, ~i, @ in standard notation.The problem of a finite basis was solved negatively in [705]. This example, which pertains to fields of characteristic 2, was later extended to the case of any prime characteristic p [67,68]. In [67] an example is also given of a (2p + 3)-dimensional Lie algebra over an infinite field of characteristic p > 0 without a finite basis of identities (in the case p = 2 there is also such an example in [705]). In [707] an example was constructed of an ...
Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings.Results in the theory of nonassociative rings and related directions reviewed in RZhMatematika during the period 1972-1978 are discussed. Special attention is given to infinite-dimensional Lie algebras, Jordan algebras, alternative rings, Mal'tsev algebras, and varieties, representations, and radicals of nonassociative rings. I. Infinite-DimensionalLie Algebras I. Books. In the Russian language there appeared during this period the successive parts of the book Lie Algebras of the series of Bourbaki [26, 27], the book of Dixmier [55], and the book of Kaplansky [91]. Two books are devoted to the theory of infinite-dimensional Lie algebras: Amayo and Stewart [240] and Bakhturin [257]. The books [365, 400, 716, 718] are written in a more traditional style. Universal enveloping algebras are also discussed in [284].2. Varieties of Lie Algebras. Although the first works properly devoted to varieties of Lie algebras appeared only in 1967 (V. A. Parfenov) and in 1968 (Yu. A. Bakhturin), the machinery and first major results of this theory actually appeared earlier in the works of A. I. Mal'tsev, A. L Kostrikin, and A. I. Shirshov. Moreover, in the hands of these specialists in this theory the theories of varieties of groups and associative algebras were advanced. The latter circumstance led to the situation that at first analogues of results in these theories were sought in varieties of Lie algebras. Thus, theorems were proved on the freeness of a semigroup of varieties of Lie algebras over a field of characteristic 0 [148], on the description of Sehreier varieties over such a field [ii], and a monotonicity theorem for verbal ideals [ii, 146, 49]. It was noted in [14] that the theorems on a semigroup of varieties and Sehreier varieties are also true in the case of an arbitrary infinitedimensional field; it is also shown there that in the case of a field of two elements the multiplication of varieties of Lie algebras is nonassociative. A description of Schreier varieties of Lie algebras, i.e., varieties in which any subalgebra of a free algebra is free, was given by M. V. Zaitsev for the case of any commutative ring: it is shown there that the Sehreier varieties are trivial; e.g., if the base ring is a field these are the varieties D, ~i, @ in standard notation.The problem of a finite basis was solved negatively in [705]. This example, which pertains to fields of characteristic 2, was later extended to the case of any prime characteristic p [67,68]. In [67] an example is also given of a (2p + 3)-dimensional Lie algebra over an infinite field of characteristic p > 0 without a finite basis of identities (in the case p = 2 there is also such an example in [705]). In [707] an example was constructed of an ...
Some properties of the right nucleus in generalized right alternative rings have been presented in this paper. In a generalized right alternative ring R which is finitely generated or free of locally nilpotent ideals, the right nucleus Nr equals the center C. Also, if R is prime and Nr ¹ C, then the associator ideal of R is locally nilpotent. Seong Nam [5] studied the properties of the right nucleus in right alternative algebra. He showed that if R is a prime right alternative algebra of char. ≠ 2 and Right nucleus Nr is not equal to the center C, then the associator ideal of R is locally nilpotent. But the problem arises when it come with the study of generalized right alternative ring as the ring dose not absorb the right alternative identity. In this paper we consider our ring to be generalized right alternative ring and try to prove the results of Seong Nam [5]. At the end of this paper we give an example to show that the generalized right alternative ring is not right alternative.
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