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$\newcommand{\iu}{{i\mkern1mu}}$Gaussian integers are complex numbers $a + b \iu$ where $a$ and $b$ are integers, and $\iu^2 = -1$.

Eisenstein integers are complex numbers $a + b \omega$ where $\omega= e^{2 \pi i/3}$, so that $\omega^3 = 1$, i.e., $\omega$ is a cube-root of unity.

Q. Are there natural generalizations of these communtative rings to $a + b \gamma$, where $a$ and $b$ are integers, and $\gamma$ is an $n$-th root of $\pm 1$?

Do they have names? Applications?

Joseph O'Rourke
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    You might be interested in cyclotomic field extensions. – Poko Sep 17 '17 at 00:30
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    According to wikipedia, one natural generalization is a Kummer ring, https://en.wikipedia.org/wiki/Kummer_ring – TomGrubb Sep 17 '17 at 00:44
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    $\mathbb{Z}[\zeta_n]$ for $n \ge 5$ is more complicated because it is dense in $\mathbb{C}$, while $\mathbb{Z}[\zeta_3]$ and $\mathbb{Z}[\zeta_4]$ are (triangular and square) lattices. – reuns Sep 17 '17 at 01:27

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The set of complex numbers of the form $a + b \gamma$, where $a$ and $b$ are integers, form a ring iff $\gamma$ is a quadratic algebraic number.

Now, $\gamma$ is an $n$-th root of unity and a quadratic algebraic number iff $n \in \{3,4,6\}$.

Thus, there is only one possible additional example: $\gamma=e^{2 \pi i/6}$, a primitive $6$-th root of unity. However, $-\gamma$ is a primitive cubic root of unity and so this ring is the same as the Eisenstein integers.

Therefore, your examples are the only possible ones.

lhf
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    It might be noteworthy that a (primitive) sixth root of unity is just the additive inverse of a primitive third root of unity. So the corresponding ring is just the ring of Eisenstein integers again. – Sebastian Schoennenbeck Dec 04 '17 at 13:28