Proposition 3

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

Algebra, chapter I, § 8. Rings

*Let $A$ be a ring, $(x_\lambda)_{\lambda \in L}$ a family of elements of $A$ and $a$ (resp. $b$ ) the set of sums $\sum_{\lambda \in L} a_\lambda x_\lambda$ where $(a_\lambda)_{\lambda \in L}$ is a family with finite support of elements of $A$ (resp. $\sum_{\lambda \in L} a_\lambda x_\lambda b_\lambda$ where $(a_\lambda)_{\lambda \in L}, (b_\lambda)_{\lambda \in L}$ are families with finite support of elements of $A$ ). Then $a$ (resp. $b$ ) is the left (resp. two-sided) ideal of $A$ generated by the elements $x_\lambda$ .

The formulae

0 = \sum_{\lambda \in L} 0.x_\lambda

\sum_{\lambda \in L} a_\lambda x_\lambda + \sum_{\lambda \in L} a'_\lambda x_\lambda = \sum_{\lambda \in L} (a_\lambda + a'_\lambda)x_\lambda

a.\sum_{\lambda \in L} a_\lambda x_\lambda = \sum_{\lambda \in L} (aa_\lambda)x_\lambda

prove that $a$ is a left ideal. Let $a'$ be a left ideal such that $x_\lambda \in a'$ for all $\lambda \in L$ and let $(a_\lambda)_{\lambda \in L}$ be a family with finite support in $A$ . Then $a_\lambda x_\lambda \in a'$ for all $\lambda \in L$ , whence $\sum_{\lambda \in L} a_\lambda x_\lambda \in a'$ ; hence $a \subset a'$ . Hence $a$ is the left ideal of $A$ generated by the $x_\lambda$ . The argument for $b$ is analogous.

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

Cited by

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Cites

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