zero divisor

Example Sentences

• "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."
• "In the ring of integers, there are no zero divisors except 0. In a ring obtained from a Boolean algebra, on the other hand, every element except the identity is a zero-divisor. The concept of a zero-divisor is intimately related to cancellation law as we see n the following proposition. 1.7 Proposition: Let R be a ring and x#92;inR. Then for all y,x#92;inR, either of the equations xy#61;xz or yx#61;zx implies y#61;z if and only if x is not a zero divisor. In other words, cancellation by an element is possible iff it is not a zero-divisor."
• "In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors."
• "If R is an integral domain, that is, has no zero divisors, then R#91;x#93; also has no zero divisors."
• "An element a of a ring is called a zero-divisor if a#92;ne 0 and ab#61;0 or ba#61;0 for some b#92;ne 0; if a is neither 0 nor a zero-divisor, it is said to be regular (see Section 7.1). A non-trivial ring without zero-divisors is called an integral domain; this term is not taken to imply commutativity."