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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 Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } of integers 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 quaternions H, {\displaystyle \mathbb {H},} which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including ± i, ± j, and ± k. In fact, the set of square roots of −1 is exactly { a i + b j + c k ∣ a 2 + b 2 + c 2 = 1 }. {\displaystyle \{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.}

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 Z / n 2 Z, {\displaystyle \mathbb {Z} /n^{2}\mathbb {Z},} any multiple of n is a square root of 0.