Let \(R\) be a commutative ring with identity, \(p \in R\) an element, and \(A\) an \(R\)-module. Show that the quotient group \(A/pA\) is naturally an \(R/(p)\)-module with scalar multiplication defined by \[ (r + (p)) \cdot (a + pA) = ra + pA. \] In particular, show that if \((p)\) is a maximal ideal, \(A/pA\) is a vector space over the field \(R/(p)\).
Solution
1. The scalar multiplication \((r + (p)) \cdot (a + pA) = ra + pA\) is well-defined.
1.1. Suppose \(r + (p) = r' + (p)\) and \(a + pA = a' + pA\).
setup of coset representatives.
1.2. Then \(r - r' = c p\) for some \(c \in R\), and \(a - a' \in pA\), so \(a - a' = p u\) for some \(u \in A\).
definition of ideals and submodules.
1.3. Compute the difference of the outputs:\[ ra - r'a' = r(a - a') + (r - r')a' = r(pu) + (cp)a' = p(ru + ca'). \]
ring and module arithmetic.
1.4. Since \(ru + ca' \in A\), \(p(ru + ca') \in pA\).
definition of \(pA = \{px : x \in A\}\).
1.5. Hence \(ra + pA = r'a' + pA\), so the action is independent of the choice of coset representatives.
3 and 4.
2. The action satisfies all module axioms:
2.1. Distributivity over module addition:\[ (r + (p)) \cdot \bigl((a + pA) + (b + pA)\bigr) = (r + (p)) \cdot (a + b + pA) = r(a+b) + pA = (ra + pA) + (rb + pA). \]
distributivity in the \(R\)-module \(A\): \(r(a+b) = ra + rb\).
2.2. Distributivity over ring addition:\[ \bigl((r + (p)) + (s + (p))\bigr) \cdot (a + pA) = (r + s + (p)) \cdot (a + pA) = (r+s)a + pA = (ra + pA) + (sa + pA). \]
module axiom \((r+s)a = ra + sa\).
2.3. Compatibility with ring multiplication:\[ \bigl((r + (p))(s + (p))\bigr) \cdot (a + pA) = (rs + (p)) \cdot (a + pA) = (rs)a + pA = r(sa) + pA = (r + (p)) \cdot (sa + pA). \]
module axiom \((rs)a = r(sa)\).
2.4. Action of the ring identity:\[ (1_R + (p)) \cdot (a + pA) = 1_R a + pA = a + pA. \]
module axiom \(1_R a = a\).
3. If \((p)\) is a maximal ideal in \(R\), then \(R/(p)\) is a field, and any module over a field is a vector space.
quotient of a commutative ring by a maximal ideal is a field.
4. Conclusion: \(A/pA\) is an \(R/(p)\)-module, and is a vector space over \(R/(p)\) when \((p)\) is maximal.
1, 2, and 3.
Q.E.D.