Let \(R\) be a ring. A left \(R\)-module is an abelian group \((M,+)\) together with a map \(R\times M\to M\), \((r,x)\mapsto rx\), such that for all \(r,s\in R\) and \(x,y\in M\):
- \(r(x+y) = rx + ry\);
- \((r+s)x = rx + sx\);
- \((rs)x= r(sx)\);
- \(1_Rx = x\).
Let \(R\) be a commutative ring. An \(R\)-algebra structure on an \(R\)-module \(M\) is an \(R\)-bilinear multiplication \(M\times M\to M\), equivalently an \(R\)-linear map \(m\colon M\otimes_R M\to M\); bilinearity means \[ r\, m(a\otimes b) = m(ra \otimes b) = m(a\otimes rb) \qquad \text{for all } r\in R \text{ and } a,b \in M. \]