Search results for "Reduced ring"
showing 4 items of 4 documents
Radical Rings with Engel Conditions
2000
Abstract An associative ring R without unity is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R ∘ under the circle operation r ∘ s = r + s + rs on R . It is proved that, for a radical ring R , the group R ∘ satisfies an n -Engel condition for some positive integer n if and only if R is m -Engel as a Lie ring for some positive integer m depending only on n .
Associative rings whose adjoint semigroup is locally nilpotent
2001
The set of all elements of an associative ring R, not necessarily with a unit element, forms a semigroup R ad under the circle operation \({r\circ s}={r+s+rs}\) on R. The ring R is called radical if R ad is a group. It is proved that the semigroup R ad is nilpotent of class n (in sense of A. Mal'cev or B. H. Neumann and T. Taylor) if and only if the ring R is Lie-nilpotent of class n. This yields a positive answer to a question posed by A. Krasil'nikov and independently considered by D. Riley and V. Tasic. It is also shown that the adjoint group of a radical ring R is locally nilpotent if and only if R is locally Lie-nilpotent.
On Associative Rings with Locally Nilpotent Adjoint Semigroup
2003
Abstract The set of all elements of an associative ring R, not necessarily with a unit element, forms a semigroup R ad under the circle operation r ∘ s = r + s + rs for all r, s in R. This semigroup is locally nilpotent if every finitely generated subsemigroup of R ad is nilpotent (in sense of A. I. Mal'cev or B. H. Neumann and T. Taylor). The ring R is locally Lie-nilpotent if every finitely generated subring of R is Lie-nilpotent. It is proved that R ad is a locally nilpotent semigroup if and only if R is a locally Lie-nilpotent ring.
Radical Rings with Soluble Adjoint Groups
2002
Abstract An associative ring R , not necessarily with an identity, is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R ∘ under the circle operation r ∘ s = r + s + rs on R . It is proved that every radical ring R whose adjoint group R ∘ is soluble must be Lie-soluble. Moreover, if the commutator factor group of R ∘ has finite torsion-free rank, then R is locally nilpotent.