Editing Square root (section)

==In rings in general==
Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring <math>\mathbb{Z}/8\mathbb{Z}</math> of integers [[modular arithmetic|modulo 8]] (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3.

Another example is provided by the ring of [[quaternion]]s <math>\mathbb{H},</math> which has no zero divisors, but is not commutative.  Here, the element −1 has [[quaternion#Square roots of −1|infinitely many square roots]], including {{math|±''i''}}, {{math|±''j''}}, and {{math|±''k''}}.  In fact, the set of square roots of {{math|−1}} is exactly<math display="block">\{ai + bj + ck \mid a^2 + b^2 + c^2 = 1\} .</math>

A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0.  For example, in <math>\mathbb{Z}/n^2\mathbb{Z},</math> any multiple of {{mvar|n}} is a square root of 0.