definition. module [jadczyk2019notes, 1.3] [ca-000X]

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Module

L∃∀N

Let \(R\) be a commutative ring. A module over \(R\), called an \(R\)-module, is a pair \((M, \bullet )\), satisfying:

1. M is a group under \(+\).
2. \(\bullet : R \to M \to M\) is called scalar multiplication, for every \(a, b \in R\), \(x, y \in M\), we have
   1. \(a \bullet (x + y) = a \bullet x + b \bullet y\)
   2. \((a + b) \bullet x = a \bullet x + b \bullet x\)
   3. \(a * (b \bullet x)=(a * b) \bullet x\)
   4. \(1_R \bullet x = x\)