Commutative algebra - Department of Mathematical Sciences - old ...

6.2. FREE MODULES 75
(2) Let K be a field and R = K[X1, X2, . . . ] the polynomial ring in countable many
variables. Show that R is a finite module, but the ideal (X1, X2, . . . ) is not a finite
module.
6.2. Free Modules
6.2.1. Definition. Let R be a ring and let R n be the free module with standard
basis e1, . . . , en.
(1) Let A = (aij) be a m × n-matrix with m rows and n columns, where the
entry aij ∈ R. Identify Rn with n-columns. Then matrix multiplication
x = (xj) ↦→ y = Ax, yi =
j
aijxj
defines a homomorphism R n → R m .
(2) Let f : R n → R m be a homomorphism. Then define a m × n-matrix A =
(aij) by
f(ej) =
i
aijei
6.2.2. Proposition. (1) The dictionary defined in 6.2.1 gives a canonical isomorphism
between the module of m × n-matrices and HomR(R n , R m ).
(2) Matrix multiplication corresponds to composition of homomorphisms and the
identity matrix corresponds to the identity homomorphism.
(3) Invertible matrices correspond to isomorphisms.
Proof. Do linear algebra homework.
6.2.3. Definition. Let R be a ring.
(1) Let A = (aij) be a m × n-matrix. The (m − 1) × (n − 1) matrix derived
from A be deleting i-row and j-column is Aij.
(2) For a square matrix A the determinant is defined by row expansion and induction:
det(a11) = a11
det A =
i
(−1) 1+j a1j det A1j
(3) The determinant of a k × k-matrix derived from A by choosing entries from
k rows and columns is a k-minor of A.
(4) If A is a n × n-matrix, then the cofactor matrix A ′ = (a ′ ij ) has entries
given by (n − 1)-minors.
a ′ ij = (−1) i+j det Aji
6.2.4. Proposition. (1) The determinant of the identity matrix is 1.
(2) The determinant is calculated by any expansion
det A =
(−1) i+j aij det Aij =
i
(3) If A, B are n × n-matrices then the product rule holds
det AB = det A det B
j
(−1) i+j aij det Aij