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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]].\nThe ring $R$ is a *finite ring* if its underlying set is finite.\nThe number of elements of $R$ is the *order* of $R$, written $\\abs{R}$.\n\n## Remarks\n\nFor each integer $n \\ge 1$, the ring $\\Z/n\\Z$ of [[integers modulo n|03VFZ2WETT]] is a finite ring of order $n$, since there are exactly $n$ residue classes.\nThe ring $M_2(\\F_2)$ of $2 \\times 2$ matrices over the [[finite field|0ZXZ82FTX2]] $\\F_2$ is a finite ring of order $2^4 = 16$, and it is not commutative.\nThe ring $\\Z$ of integers is not a finite ring, because it has infinitely many elements.\nEvery finite [[integral domain|0QVD7WBFGN]] is a field: multiplication by a nonzero element is injective, hence bijective on a finite set, so every nonzero element has an inverse.\n",
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