tensor products of algebras over a commutative ring

LEMMA 6.3   Let $A$ be a commutative ring. Let $B,C$ be $A$ -algebras. Then the followings are true.

1. The module
   $$B\otimes_A C$$
   carries a natural structure of $A$ -algebra.
2. There exits $A$ -algebra homomorphisms
   $$\iota_B: B \ni b \mapsto b\otimes 1 \in B\otimes_A C,\qquad \iota_C: C \ni c \mapsto 1\otimes c \in B\otimes_A C.$$

3. The triple $(B\otimes_A C, \iota_B,\iota C)$ has the following universal property: For any $A$ -algebra $D$ and for any $A$ -algebra homomorphisms $f:B\to D$ and $g:C\to D$ , there exists a unique $A$ -algebra homomorphism
   $$h:B\otimes_A C \to D$$
   such that $f=h\circ \iota_B$ and $g=h \circ \iota_C$ .

PROOF.. An easy exercise.

$\qedsymbol$

Note that in the situation of the above Lemma, if $A$ is non commutative, then $B\otimes_A C$ may not have a natural structure of ring. Sooner or later one needs to face this fact.