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  "body": "\n## Definition\n\nLet $K$ be a commutative ring and let $A$ be an [[associative algebra|1TZSV19D85]] over $K$ with unit element $1$.\nA *subalgebra* of $A$ is a subset $B \\subseteq A$ such that:\n\n1. $1 \\in B$.\n2. If $a, b \\in B$, then $a + b \\in B$ and $ab \\in B$.\n3. If $k \\in K$ and $b \\in B$, then $kb \\in B$.\n\n## Remarks\n\nClause 3 with $k = -1$ gives $-b \\in B$, so $B$ is a [[subring|0859D58WPD]] of $A$ and a [[submodule|0RGZ10DFPG]] of the $K$-module $A$.\nWith the operations of $A$, the set $B$ is then a $K$-algebra.\nLet $\\eta \\colon K \\to A$, $k \\mapsto k1$, be the structure map.\nA subset $B$ is a subalgebra exactly when it is a subring of $A$ that contains $\\eta(K)$, because $kb = \\eta(k)b$.\n\nExample: in the $\\R$-algebra $M_2(\\R)$ of real $2 \\times 2$ matrices, the set of upper triangular matrices is a subalgebra.\nThe identity matrix is upper triangular, and sums and real multiples of upper triangular matrices are upper triangular.\nFor upper triangular $P$ and $Q$, the $(2,1)$ entry of $PQ$ is $p_{21} q_{11} + p_{22} q_{21} = 0$, since $p_{21} = q_{21} = 0$.\n\nNon-example: in the $\\R$-algebra $\\R[x]$, the set $\\Z[x]$ of polynomials with integer coefficients satisfies clauses 1 and 2 but not clause 3.\nFor $k = 1/2$ and $b = 1$ the product $kb = 1/2$ is not in $\\Z[x]$.\nSo $\\Z[x]$ is a subring of $\\R[x]$ but not an $\\R$-subalgebra.\n\nSome authors work with algebras without a unit element and drop clause 1.\nFor a field $K$, a subalgebra is a linear subspace of $A$ that contains $1$ and is closed under multiplication.\n",
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