Near-rings in which each prime factor is simple.
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Near-rings with a special condition on idempotents.
Heatherly, H.E., Courville, J.R. (1999)
Mathematica Pannonica
New characterizations of von Neumann regular rings and a conjecture of Shamsuddin.
Carl Faith (1996)
Publicacions Matemàtiques
A theorem of Utumi states that if R is a right self-injective ring such that every maximal ideal has nonzero annihilator, then R modulo the Jacobson radical J is a finite product of simple rings and is a von Neuman regular ring. We prove two theorems and a conjecture of Shamsuddin that characterize when R itself is a von Neumann ring, using a splitting theorem of the author on when the maximal regular ideal of a ring splits off.
Nil-extensions of completely simple semirings
Sunil K. Maity, Rituparna Ghosh (2013)
Discussiones Mathematicae - General Algebra and Applications
A semiring S is said to be a quasi completely regular semiring if for any a ∈ S there exists a positive integer n such that na is completely regular. The present paper is devoted to the study of completely Archimedean semirings. We show that a semiring S is a completely Archimedean semiring if and only if it is a nil-extension of a completely simple semiring. This result extends the crucial structure theorem of completely Archimedean semigroup.
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