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  "body": "\n## Definition\n\nLet $I$ be a set and let $(R_i)_{i \\in I}$ be a family of [[rings|0894J980RK]].\nThe product $\\prod_{i \\in I} R_i$ is the set of families $(a_i)_{i \\in I}$ with $a_i \\in R_i$ for each $i \\in I$.\nIts addition is $(a_i)_{i \\in I} + (b_i)_{i \\in I} = (a_i + b_i)_{i \\in I}$.\nIts multiplication is $(a_i)_{i \\in I} \\cdot (b_i)_{i \\in I} = (a_i \\cdot b_i)_{i \\in I}$.\nWith these operations, $\\prod_{i \\in I} R_i$ is a ring, with zero $(0)_{i \\in I}$ and identity $(1)_{i \\in I}$.\n\n## Remarks\n\nFor two rings $R$ and $S$, the product is $R \\times S = \\{(r, s) : r \\in R, s \\in S\\}$.\nFor each $j \\in I$, the projection $\\prod_{i \\in I} R_i \\to R_j$, $(a_i)_{i \\in I} \\mapsto a_j$, is a [[ring homomorphism|0GGE0Z0JQ1]].\nThe product of two nonzero rings is not an [[integral domain|09S3PFBEPM]]: in $R \\times S$, $(1, 0) \\cdot (0, 1) = (0, 0)$.\n\n## Explanations\n\n### L1 human\n\nTo multiply two pairs, you multiply the first entries and the second entries apart.\nFor example, in $\\mathbb{Z} \\times \\mathbb{Z}$, $(2, 3) \\cdot (4, 5) = (8, 15)$.\n",
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