Definition 3

Tag 05H9, permanent. Cite it as [Bourbaki, Tag 05H9].

Algebra, chapter I, § 8. Rings

Let $A$ be a ring. A subring of $A$ is any subset $B$ of $A$ which is a subgroup of $A$ under addition, which is stable under multiplication and which contains the unit of $A$ .

The above conditions may be written as follows

0 \in B, \quad B + B \subset B, \quad -B \subset B, \quad B.B \subset B, \quad 1 \in B.

If $B$ is a subring of $A$ , it is given the addition and multiplication induced by those on $A$ , which make it into a ring. The canonical injection of $B$ into $A$ is a ring homomorphism.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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