Properties
In the ring of n-by-n matrices over some field, the left and right zero divisors coincide; they are precisely the non-zero singular matrices. In the ring of n-by-n matrices over some integral domain, the zero divisors are precisely the non-zero matrices with determinant zero.
Left or right zero divisors can never be units, because if a is invertible and a b = 0, then 0 = a−10 = a−1a b = b.
Every non-trivial idempotent ring element a is a zero divisor, since a2 = a implies that a (a − 1) = (a − 1) a = 0, with nontriviality ensuring that neither factor is 0. Nonzero nilpotent ring elements are also trivially zero divisors.
The set of zero divisors is the union of the associated prime ideals of the ring.
Read more about this topic: Zero Divisor
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)