zero divisor

Definitions

1. 1

   An element a of a ring R for which there exists some nonzero element x ∈ R such that…

   An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

   • An idempotent element e#92;ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)#61;0#61;(1-e)e.
   • Linnell [25, 1977] proved that if G is a torsion-free abelian by locally finite by super-solvable group and K is any field, then K[G] has no nontrivial zero divisors.
2. 2

   A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such…

   A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

   • If R is an integral domain, that is, has no zero divisors, then R#91;x#93; also has no zero divisors.

The neighborhood

• antonymregular elementantonym(s) of “any element whose product with some nonzero element is zero”
• neighborannihilator
• neighborintegral domain
• neighbornilpotent
• neighbortrivial zero divisorany element whose product with some nonzero element is zero
• neighborexact zero divisorboth senses
• neighborleft zero divisorboth senses
• neighborright zero divisorboth senses
• neighbortwo-sided zero divisorboth senses