Editing Division by zero (section)

===Abstract algebra===
In [[abstract algebra]], the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a [[commutative ring]], which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called [[Localization (commutative algebra)|localization]]. However, the localization of every commutative ring at zero is the [[trivial ring]], where <math>0 = 1</math>, so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.

Nevertheless, any number system that forms a [[commutative ring]] can be extended to a structure called a [[Wheel theory|wheel]] in which division by zero is always possible.<ref>{{citation |year=2004 |last=Carlström |first=Jesper |title=Wheels: On Division by Zero |journal=Mathematical Structures in Computer Science |doi=10.1017/S0960129503004110 |volume=14 |issue=1 |pages=143–184 |doi-broken-date=1 November 2024 |url=http://www2.math.su.se/reports/2001/11/ }}</ref> However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element <math>1</math>, and if the original system was an [[integral domain]], the multiplication in the wheel no longer results in a [[cancellative semigroup]].

The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as [[Ring (mathematics)|rings]] and [[Field (mathematics)|fields]]. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a [[skew field]] (which for this reason is called a [[division ring]]). However, in other rings, division by nonzero elements may also pose problems. For example, the ring '''Z'''/6'''Z''' of integers mod 6. The meaning of the expression <math display="inline">\frac{2}{2}</math> should be the solution ''x'' of the equation <math>2x = 2</math>. But in the ring '''Z'''/6'''Z''', 2 is a [[zero divisor]]. This equation has two distinct solutions, {{math|1=''x'' = 1}} and {{math|1=''x'' = 4}}, so the expression <math display="inline">\frac{2}{2}</math> is [[Defined and undefined|undefined]].

In field theory, the expression <math display="inline">\frac{a}{b}</math> is only shorthand for the formal expression ''ab''<sup>−1</sup>, where ''b''<sup>−1</sup> is the multiplicative inverse of ''b''. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when ''b'' is zero. Modern texts, that define fields as a special type of ring, include the axiom {{math|0 ≠ 1}} for fields (or its equivalent) so that the [[zero ring]] is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.