Endomorphisms of free modules as sums of four quadratic endomorphisms

Breaz, Simion

doi:10.1080/03081087.2017.1389853

Cited by 5 publications

(1 citation statement)

References 13 publications

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“…In particular, Harris proved that if the additive group of a division ring D is generated by commutators then every quadratic matrix over D is a sum on nilpotent elements, [11]. The techniques from this paper were used in [4] to prove that every endomorphism of an infinitely generated free module is a sum of four square-zero endomorphisms. For other results about sums of square-zero endomorphisms we refer to [17] and [20].…”

Section: Introductionmentioning

confidence: 99%

When is every non central-unit a sum of two nilpotents?

Breaz¹,

Zhou²

2021

Preprint

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Motivated by the notions of 2-good rings and fine rings, we define a 2-nilgood ring as a ring whose every non central-unit is the sum of two nilpotents. This is a study of 2-nilgood rings. The following results are obtained: (a) a ring is 2-nilgood iff it is either a 2-nilgood simple ring or a commutative local ring with nil Jacobson radical; (b) a 2-nilgood simple ring is a field if it is right Goldie; (c) a 2-nilgood ring containing a uniform right ideal is a commutative local ring with nil Jacobson radical; (d) a ring whose every non central-unit is a sum of a square-zero element and a nilpotent is a commutative local ring with nil Jacobson radical. We conclude with the conjecture that a 2-nilgood simple ring is a field.

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