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The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$

Statement

Let $R$ be a unital ring. For every left $R$ -module $N$ and every right $R$ -module $M$ , the maps

$λ_{N} : R \otimes_{R} N ⟶ N, r \otimes n ⟼ r n,$

and

$ρ_{M} : M \otimes_{R} R ⟶ M, m \otimes r ⟼ m r,$

are group isomorphisms. Their inverses are $n \mapsto 1_{R} \otimes n$ and $m \mapsto m \otimes 1_{R}$ , respectively. The maps respect every displayed outer module structure.

Facts & Assumptions

Given: A unital ring $R$ , a left $R$ -module $N$ , and a right $R$ -module $M$ .

[L1]

Balanced pairings induce unique homomorphisms from tensor products (Universal property of the tensor product for balanced maps into abelian groups).

[L2]

An elementary-tensor prescription descends exactly when the corresponding pairing is balanced (A formula on elementary tensors defines a homomorphism exactly when its underlying pairing is balanced).

Proof

technique · direct

1.1givenL1L2

The pairing $(r, n) \mapsto r n$ is additive in each variable and satisfies $(r s) n = r (s n)$ , so it is balanced; [L1] and [L2] give $λ_{N}$ .

1.2L1L2algebra

The pairing $(m, r) \mapsto m r$ is balanced, so [L1] and [L2] give $ρ_{M}$ . Define the additive map $η_{M} (m) = m \otimes 1_{R}$ . Then $ρ_{M} η_{M} (m) = m$ , while $η_{M} ρ_{M} (m \otimes r) = m r \otimes 1_{R} = m \otimes r$ by balance; uniqueness in [L1] makes the second composite the identity. Thus $ρ_{M}$ and $η_{M}$ are inverse.

2.1step 1.1L1algebra

The additive map $η_{N} : N \to R \otimes_{R} N$ , $n \mapsto 1_{R} \otimes n$ , satisfies $λ_{N} η_{N} (n) = n$ , while $η_{N} λ_{N} (r \otimes n) = 1_{R} \otimes r n = r \otimes n$ by balance. Uniqueness in [L1] makes the second composite the identity, so $λ_{N}$ and $η_{N}$ are inverse.

3.1step 1.1step 2.1step 1.2algebra

Each map commutes with any outer scalar action by associativity of that action, checked on elementary tensors. The calculations remain valid for the zero ring and for zero modules, where all displayed maps are the unique maps between zero groups.

4.1step 2.1step 1.2step 3.1∎

Therefore the regular module is a left and right tensor unit with the stated natural formulas.

Depends on

Dependency tree · 2 levels

Sources

Stacks Project, Section 10.12: Tensor products