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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.
But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression is undefined. In field theory, the expression is only shorthand for the formal expression ab −1, where b −1 is the multiplicative inverse of b.
A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in ...
Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a ...
The zero divisor problem asks whether this is the only obstruction; in other words, Given a field K and a torsion-free group G , is it true that K [ G ] contains no zero divisors? No counterexamples are known, but the problem remains open in general (as of 2017).
Let K be a field, and G a torsion-free group. 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 ...
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]
The element 0 in the zero ring is not a zero divisor. The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither maximal nor prime. The zero ring is generally excluded from fields, while occasionally called as the trivial field. Excluding it agrees with the fact that its ...