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An element that is a left or a right zero divisor is simply called a zero divisor. [2] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.
In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general example can be written as a 0 {\displaystyle {\tfrac {a}{0}}} , where a {\displaystyle a} is the dividend (numerator).
The graph of all zero divisors is non-empty for every ring that is not an integral domain. It remains connected, has diameter at most three, [3] and (if it contains a cycle) has girth at most four. [4] [5] The zero-divisor graph of a ring that is not an integral domain is finite if and only if the ring is finite. [3]
If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0. [2]
e i e j = −e j e i ≠ e j e i if i, j are distinct and non-zero, nor associative: (e i e j) e k = −e i (e j e k) ≠ e i (e j e k) if i, j, k are distinct, non-zero and e i e j ≠ ±e k. The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative.
Kaplansky's zero divisor conjecture states: The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are known as, respectively, Kaplansky's idempotent conjecture: K[G] does not contain any non-trivial idempotents, i.e., if a 2 = a, then a = 1 or a = 0.
More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero-divisor. On the other hand, not every reduced ring is an integral domain; for example, the ring Z [ x , y ]/( xy ) contains x + ( xy ) and y + ( xy ) as zero-divisors, but no non-zero nilpotent elements.
It is sometimes required that r is not a zero-divisor, and some authors [10] require that R is a domain.) For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M. [11] [12] (This type of divisible module is also called principally injective module.)