Local rings whose multiplicative group is nilpotent

Catino, Francesco; Miccoli, Maria Maddalena

doi:10.1007/s00013-003-4672-6

Cited by 5 publications

(6 citation statements)

References 6 publications

(6 reference statements)

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“…The validity of assertion (i) for nilpotent and local rings was established in [8] and [9], respectively. Consequently, if R has decomposition (3.1), then Z n ( R i ) ° = Z R n i (°) for all n ≥ 0.…”

Section: Example 31 An Artinian Ring R That Has a Nilpotent Adjointmentioning

confidence: 98%

“…This conjecture was confirmed by Du in [8], where the following result was obtained: In a radical ring R, each term Z n ( R ) ( n ≥ 0 ) of its upper central series coincides with the corresponding term Z n ( R° ) ( n ≥ 0 ) of the upper central series of the group R°. In the case where the ring R is local, the corresponding result was obtained in [9] and is formulated as follows: Z n ( R ) * = Z n ( R * ) for every n > 0; here, Z n ( R * ) is the n th term of the upper central series of the group R * . In particular, the multiplicative group R * of a local ring R is nilpotent if and only if the ring R is Lie-nilpotent and the nilpotency classes coincide for both structures.…”

Section: Theorem B the Adjoint Group R° Of A Finite Ring R Is Nilpotmentioning

confidence: 99%

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Artinian rings with nilpotent adjoint group

Evstaf’ev

2006

Ukr Math J

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Let R be an Artinian ring (not necessarily with unit element), let Z ( R ) be its center, and let R°b e the group of invertible elements of the ring R with respect to the operation a ° b = a + b + a b. We prove that the adjoint group R° is nilpotent and the set Z ( R ) + R° generates R as a ring if and only if R is the direct sum of finitely many ideals each of which is either a nilpotent ring or a local ring with nilpotent multiplicative group.

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Section: Example 31 An Artinian Ring R That Has a Nilpotent Adjointmentioning

confidence: 98%

Section: Theorem B the Adjoint Group R° Of A Finite Ring R Is Nilpotmentioning

confidence: 99%

Artinian rings with nilpotent adjoint group

Evstaf’ev

2006

Ukr Math J

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“…By the same argument, if K := J(R) + ⟨a i , a j ⟩ rg for any positive integers i and j, then K ′ is finite and thus it is central in R. Hence [R, R] ⊆ Z(R) and therefore R is Lie nilpotent. The unit group U (R) is also nilpotent by [14,Corollary 3.4] what means that…”

Section: Unitary Ringsmentioning

confidence: 99%

Minimal non-BFC rings

2023

Algebra and Discrete Mathematics

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“…It was recently shown by Catino and Miccoli [6] and independently by Stolz [10] that a local ring R is Lie nilpotent if and only if its multiplicative group R * is nilpotent and that the classes of nilpotency of both structures coincide. Furthermore, it is obvious that a local ring is 1-Engel as a Lie ring if and only if its multiplicative group is 1-Engel.…”

Section: Introductionmentioning

confidence: 99%

Semilocal rings with n-Engel multiplicative group

Amberg

Sysak

2004

Arch. Math.

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An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R * of a semilocal ring R generated by 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. Introduction.Let R be an associative ring with a unit element 1. The group of all invertible elements of R is called the multiplicative group of R and is denoted by R * . The Jacobson radical of R is its unique ideal J which is maximal with respect to the condition that the set 1 + J is a normal subgroup of R * . It is easy to see that the factor group R * /(1 + J ) coincides with the multiplicative group (R/J ) * of R/J . The ring R is artinian if the minimal condition for the one-sided ideals of R is satisfied, and R is a division ring if every non-zero element of R is invertible. The ring R is local if the factor ring R/J is a division ring and R is semilocal if R/J is an artinian ring. It is well known that in the latter case R/J is a direct sum of rings each of which is isomorphic to a ring of square matrices over a division ring.Recall that every ring R may be considered as a Lie ring under the Lie multiplication [r, s] = rs − sr for all r, s ∈ R. If r 1 , r 2 , . . . are elements of R, the Lie-commutators [r 1 , . . . , r n+1 ] are defined inductively by [r 1 , . . . , r n+1 ] = [[r 1 , . . . , r n ], r n+1 ] for all n 2. The ring R is called Lie nilpotent of class n, if [r 1 , . . . , r n+1 ] = 0 for all r 1 , . . . r n+1 of R and n is the least integer with this property. Moreover, R is Engel if [r, s, . . . , s] = 0 for each pair of elements r and s of R, and n-Engel if s appears exactly n times. Also, R is a locally Lie nilpotent ring if every finitely generated subring of R is Lie nilpotent. Obviously in this case R is locally nilpotent as a Lie ring. Note that nilpotent, Engel and n-Engel groups are defined in a corresponding way where the usual group commutator replaces the

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Local rings whose multiplicative group is nilpotent

Abstract: We prove that the upper central chain of the multiplicative group of a local ring R coincides with the chain of the multiplicative group of terms of the upper central chain of the associated Lie ring of R.

Cited by 5 publications

References 6 publications

Artinian rings with nilpotent adjoint group

Artinian rings with nilpotent adjoint group

Minimal non-BFC rings

Semilocal rings with n-Engel multiplicative group

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