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Algebra**

An an algebra \((\mathcal{A},\diamond)\) over a ring \(R\) is an \(R\)-module together with an \(R\)-bilinear binary product \(\diamond:\mathcal{A}\times\mathcal{A}\to\mathcal{A}\) - i.e. \(forall x,y,z,w\in \mathcal{A}:\forall r,s\in R: (r\cdot x+y)\diamond(s\cdot z+w)=(rs)\cdot(x\diamond z)+r\cdot(x\diamond w)+s\cdot(y+z)+y\diamond w\). Typically one specifies the case where \(R\) is a field and \(\diamond\) is associative.