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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.
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 ...
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 ...
All nilpotent elements are zero divisors. An n × n {\displaystyle n\times n} matrix A {\displaystyle A} with entries from a field is nilpotent if and only if its characteristic polynomial is t n {\displaystyle t^{n}} .
Whether this is true if R is a field of positive characteristic remains unknown. A long-standing conjecture of Kaplansky (~1940) says that if G is a torsion-free group, and K is a field, then the group ring K[G] has no non-trivial zero divisors. This conjecture is equivalent to K[G] having no non-trivial nilpotents under the same hypotheses for ...
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.
An element r of R is a called a two-sided zero divisor if it is both a left zero divisor and a right zero divisor. division A division ring or skew field is a ring in which every nonzero element is a unit and 1 ≠ 0. domain A domain is a nonzero ring with no zero divisors except 0.
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 ...