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Subalgebra**

Given an algebra \((\mathfrak{A},\diamond)\) over a ring \(R\), a subalgebra of \(\mathfrak{A}\) is a nonempty subset \(\mathfrak{B}\subset\mathfrak{A}\) such that \((\mathfrak{B},\diamond)\) is also an algebra, i.e. \(\forall x,y\in\mathfrak{B}:\forall r\in R: x+y\in\mathfrak{B}\land rx\in \mathfrak{B}\land -x\in\mathfrak{B}\land x\diamond y\in\mathfrak{B}\).