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  "body": "\n## Summary\n\nLet $A$ be an [[associative algebra|3CTW1P4HBP]] over a commutative ring $K$.\nA subalgebra of $A$ is a subset that contains $1$ and is itself a $K$-algebra under the same operations.\nIt is a [[subring|0GNCRPYXYV]] of $A$ that is also a [[submodule|05WDSN597C]] for the scalar multiplication by $K$.\nSubalgebras are the natural subobjects of algebras.\nImages of algebra homomorphisms are subalgebras, and so are the [[subalgebras generated|2TQD8NMG9P]] by a few elements, such as $K[a] \\subseteq A$.\nIt is the [[substructure|676H7EGTKJ]] for the operations of a $K$-algebra, with $1$ as a constant.\n\n## Explanations\n\n### L1 human\n\nTake an algebra, a place where you can add, multiply and scale by numbers.\nA subalgebra is a smaller piece that you never leave when you do these operations, and that contains $1$.\nFor example, in the algebra of all $2 \\times 2$ real matrices, the upper triangular matrices form a subalgebra.\nAdding, multiplying or scaling upper triangular matrices gives an upper triangular matrix, and the identity matrix is upper triangular.\nThe polynomials with integer coefficients do not form a subalgebra of the real polynomials $\\R[x]$, because scaling $1$ by $1/2$ leaves the set.\n",
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