(1)Let A be a ring

Let A be a ring.

1. Tensor Products

Let M and N be right and left A-modules. A middle linear map f from M × N to an abelian group C is a function f : M × N → C such that for all mi, m ∈ M and ni, n ∈ N and a ∈ A,

(1) f (m₁+ m₂, n) = f (m₁, n) + f (m₂, n) (2) f (m, n1+ n2) = f (m, n1) + f (m, n2) (3) f (mr, n) = f (m, rn).

We shall denote M(M, N ) the category of middle linear maps on M × N. This category is defined as follows. Objects of M(M, N ) are middle linear maps f : M × N → C, where C is an abelian group. A morphism from f : M × N → C to g : M × N → D is a group homomorphism h : C → D such that g = h ◦ f.

Definition 1.1. The tensor product of M and N is a universal object in the category M(M, N ). (It is defined unique up to isomorphisms)

Theorem 1.1. The tensor product of M and N always exists.

Let F be the free abelian group generated by the set M × N and K be its subgroup generated by all elements of the form

(1) (m + m⁰, n) − (m, n) − (m⁰, n) (2) (m, n + n⁰) − (m, n) − (m, n⁰), (3) (ma, n) − (m, an).

We denote the quotient group F/K by M ⊗AN. The coset (m, n) + K is denoted by m ⊗ n.

The quotient map is denoted by π : F → M ⊗_AN. Let i : M × N → F be the inclusion map. Set ι = π ◦ i. By definition, we know that ι : M × N → M ⊗AN is a middle linear map on M × N.

Let f : M × N → C be any other middle linear map. We define f⁰ : M ⊗_AN → C

by f⁰(m ⊗ n) = f (m, n) and extend f additively. Then we can check that f⁰ is well-defined and hence by definition, f⁰◦ ι = f. Now, we will prove that such f⁰ is unique. Suppose that g : M ⊗_AN → C is another homomorphism so that g ◦ ι = f. Then g(m ⊗ n) = f (m, n) = f⁰(m ⊗ n). Then g = f⁰ on the set of generators of M ⊗AN and hence g = f on M ⊗AN.

We conclude that ι : M × N → M ⊗_AN is the tensor product of M and N.

1.1. Tensor product of modules over commutative rings. When A is a commutative ring, we can consider the category of bilinear maps on M × N. A bilinear map is a function f : M × N → Z from M × N to an A-module Z such that for m, m_i∈ M, n, n_i∈ N, a ∈ A :

(1) f (m1+ m2, n) = f (m1, n) + f (m2, n) (2) f (m, n₁+ n₂) = f (m, n₁) + f (m, n₂) (3) f (am, n) = f (m, an) = af (m, n).

Definition 1.2. The tensor product of M and N is the universal object in the category of bilinear maps on M × N.

Theorem 1.2. The tensor product of M and N exists.

Let F be the free A-module generated by M × N and K be the A-submodule generated by elements of the form

1

2

(1) (m + m⁰, n) − (m, n) − (m⁰, n) (2) (m, n + n⁰) − (m, n) − (m, n⁰), (3) (am, n) − a(m, n).

(4) (m, an) − a(m, n).

The quotient group F/K denoted by M ⊗AN is an A-module. The coset (m, n) + K is also denoted by m ⊗ n.We let ι : M × N → M ⊗_AN be the natural map: the composition of M × N → F → M ⊗A N. Given a bilinear map f : M × N → Z, we can define f⁰ : M ⊗_AN → Z by f⁰(m ⊗ n) = f (m, n). It is easy to verify ι : M × N → M ⊗_AN is the tensor product of M and N.