ENUMERATION OF IDEALS OF SOME NILPOTENT MATRIX RINGS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250140
Author(s):  
GEORGY P. EGORYCHEV ◽  
FERIDE KUZUCUOĞLU ◽  
VLADIMIR M. LEVCHUK
Keyword(s):  
Integral Representation ◽  
Matrix Rings ◽  
Nilpotent Matrix ◽  
Combinatorial Sums

Using the method of integral representation of combinatorial sums we enumerate ideals of certain nilpotent matrix rings.

Fine rings: A new class of simple rings

2016 ◽  
Vol 15 (09) ◽  
pp. 1650173 ◽  
Author(s):  
G. Cǎlugǎreanu ◽  
T. Y. Lam
Keyword(s):  
Division Ring ◽  
Nilpotent Element ◽  
Nonzero Element ◽  
Proper Class ◽  
Invertible Matrix ◽  
Matrix Rings ◽  
Artinian Rings ◽  
New Class ◽  
Nilpotent Matrix ◽  
Square Matrix

A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly the class of all simple artinian rings. One of the main results in this paper is that matrix rings over fine rings are always fine rings. This implies, in particular, that any nonzero (square) matrix over a division ring is the sum of an invertible matrix and a nilpotent matrix.

On the Generalized Strongly Nil-Clean Property of Matrix Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 625-634
Author(s):  
Aleksandra S. Kostić ◽  
Zoran Z. Petrović ◽  
Zoran S. Pucanović ◽  
Maja Roslavcev
Keyword(s):  
Matrix Rings ◽  
Nilpotent Matrix ◽  
Unital Ring ◽  
Sum Formula ◽  
Matrix Formula

Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].

DERIVATIONS OF THE LOCALLY NILPOTENT MATRIX RINGS

2010 ◽  
Vol 09 (05) ◽  
pp. 717-724 ◽  
Author(s):  
VLADIMIR M. LEVCHUK ◽  
OKSANA V. RADCHENKO
Keyword(s):  
Associative Ring ◽  
Jordan Derivation ◽  
Matrix Rings ◽  
Nilpotent Matrix ◽  
Locally Nilpotent

Derivations of the ring of all finitary niltriangular matrices over an arbitrary associative ring with identity for any chain of matrix indices are described. Every Lie or Jordan derivation is a derivation of this ring modulo third hypercenter.

scholarly journals Integral Representation and the Computation of Multiple Combinatorial Sums from Hall’s Commutator Theory

2021 ◽  
pp. 12-20
Author(s):  
Georgy P. Egorychev ◽  
Sergey G. Kolesnikov ◽  
Vladimir M. Leontiev
Keyword(s):  
Integral Representation ◽  
Closed Form ◽  
Combinatorial Identities ◽  
Chevalley Groups ◽  
Sylow Subgroups ◽  
Commutator Theory ◽  
Combinatorial Sums

In this paper we prove a series of combinatorial identities arising from computing the exponents of the commutators in P. Hall’s collection formula. We also compute a sum in closed form that arises from using the collection formula in Chevalley groups for solving B. A. F. Wehrfritz problem on the regularity of their Sylow subgroups

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