I am dealing with two differents definitions of ideals. Given a subset $S$ of a commutative ring $A$, the ideal generated by $S$, called $\langle S \rangle $ is:
- The smallest ideal of $A$ that contains $S$;
- The set of the $A$-finite combinations $\{\sum_{i=1}^n a_is_i; n\in\mathbb{N}, a_i\in A, s_i\in S\}$.
My doubt is the following: can I prove that $S\subseteq\langle S\rangle $ in the second definition without the statement of $A$ has unit? I think no, for instance: $A=2\mathbb{Z}$ and $S=\{2\}$. At the second definition I'll have $\langle S\rangle=4\mathbb{Z}$...? It looks strange for me.
Many thanks.
The multiplication by $n$ is defined in any abelian group no need that $A$ is unital then $$\langle S\rangle\ =\ \{ \sum_{i=1}^I (a_i s_i+n_is_i), a_i \in A,n_i \in \Bbb{Z}, s_i \in S\}$$ is the smallest ideal of $A$ containing $S$