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  "body": "\n## Summary\n\nAn associative algebra over a commutative ring $K$ is a [[ring|0B1261B5ED]] whose elements can also be multiplied by scalars from $K$.\nThe scalar product is compatible with the ring product.\nMatrix algebras, polynomial algebras, group algebras and endomorphism algebras of vector spaces are the standard examples.\nThe notion puts ring theory and linear algebra in one frame, so that dimension and bases are available when $K$ is a field.\nA [[commutative algebra over a ring|DPBJ04JMH8]] is the special case where the ring product is commutative.\n\n## Explanations\n\n### L1 human\n\nAn algebra is a vector space, or more generally a module, in which vectors can also be multiplied with each other.\nThe multiplication is associative and works well with scaling: scaling a product is the same as scaling one factor.\nThe $n \\times n$ real matrices form an algebra over $\\R$: they can be added, multiplied, and scaled by real numbers.\nThe complex numbers are a $2$-dimensional algebra over $\\R$, with basis $1, i$ and the rule $i^2 = -1$.\n",
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