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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nThe module $M$ is *uniform* if:\n\n1. $M \\ne 0$;\n2. $N_1 \\cap N_2 \\ne 0$ for all nonzero [[submodules|0RGZ10DFPG]] $N_1, N_2$ of $M$.\n\n## Remarks\n\nEquivalently, $M \\ne 0$ and every nonzero submodule of $M$ is [[essential|8MS51FASMX]] in $M$.\nExample: the $\\Z$-module $\\Z$ is uniform.\nThe check: nonzero submodules $a\\Z$ and $b\\Z$ both contain $ab \\ne 0$.\nExample: $\\Z/p^n\\Z$, for $p$ prime and $n \\ge 1$, is uniform, since its subgroups form a chain.\nNon-example: $\\Z \\oplus \\Z$ fails clause 2, since $\\Z \\oplus 0$ and $0 \\oplus \\Z$ meet only in $0$.\nEvery uniform module is [[indecomposable|E2P29GCD9T]].\nThe converse fails: let $K$ be a field and $A = K[x, y]/(x^2, xy, y^2)$.\nAs a module over itself, $A$ is indecomposable, because $A$ is a commutative local ring and so has no idempotents other than $0$ and $1$.\nIt is not uniform, since its submodules $Ax = Kx$ and $Ay = Ky$ meet only in $0$.\n",
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