Let m,n be positive integers, v a multilinear commutator word and w = vm. We prove that if G is a residually finite group in which all wvalues are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover...

Let B be a commutative $\mathbb {Z}$ -graded domain of characteristic zero. An element f is said to cylindrical if it nonzero, homogeneous nonzero degree, and such that B(f) polynomial ring in one variable over subring. We study the relation between existence locally nilpotent derivation B. Also, given d ≥ 1, we give sufficient conditions guarantee every $B^{(d)} = {\bigoplus }_{i \in \mathbb {...

LOCALLY GRADED GROUPS WITH CERTAIN MINIMAL CONDITIONS FOR SUBGROUPS, II JAVIER OTAL AND JUAN MANUEL PEÑA This paper deals with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions . If P is a theoretical property of groups and subgroups, we show that a locally graded group G satisfies the minimal condition...

A result of Artin, Small, and Zhang is used to show that a noetherian algebra over a commutative, noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, noetherian associated graded ring. This result is extended to show that if an algebra over a commutative noetherian ring has a locally finite, noetherian associated graded ring, then the intersection of the powers ...

Some locally finite simple Lie algebras are graded by finite (possibly nonreduced) root systems. Many more algebras are sufficiently close to being root graded that they still can be handled by the techniques from that area. In this paper we single out such Lie algebras, describe them, and suggest some applications of such descriptions.

For any non-torsion group G with identity e, we construct a strongly G-graded ring R such that the Jacobson radical J(Re) is locally nilpotent, but J(R) is not locally nilpotent. This answers a question posed by Puczy lowski.

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite ...