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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 zero-divisor graph of , the only possible zero-divisor graph that is a tree but not a star. In mathematics, and more specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring.
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]
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.
However, zero is its own greatest divisor if greatest is understood in the context of the divisibility relation, so gcd(0, 0) is commonly defined as 0. This preserves the usual identities for GCD, and in particular Bézout's identity , namely that gcd( a , b ) generates the same ideal as { a , b } .
Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches: Say that 26 cannot be divided by 11; division becomes a partial function.
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]