Given a noncommutative nonunital ring $R$ and a subset $S\subseteq R$, I know that the left and right ideals generated by $S$ in $R$ have the forms$$RS:=\{\sum^n_{i=1} r_is_i:n\in\mathbb{N}\backslash\{0\},r_i\in R,s_i\in S \}$$and$$SR:=\{\sum^n_{i=1}s_ir_i:n\in\mathbb{N}\backslash\{0\},r_i\in R,s_i\in S \}$$respectively.
I also know that the two sided ideal generated by $S$ in $R$ has the form$$RSR:=\{\sum^n_{i=1}r_is_i\tilde{r}_i:n\in\mathbb{N}\backslash\{0\},r_i,\tilde{r}_i\in R,s_i\in S \}.$$
Now, my question is, how do the above ideals differ from a general ideal generated by $S$ in $R$?
For example, maybe $R=x\mathbb{C}\langle x,y\rangle$ is the free algebra over $\mathbb{C}$ whose basis consists of words in $x$ and $y$ beginning with $x$ on the left and whose multiplication is concatenation of words.
Take $S=x\mathbb{C}\langle x\rangle=x\mathbb{C}[x]$ to be the set of nonconstant polynomials over $\mathbb{C}$.
Then, clearly none of $RS$, $SR$ or $RSR$ contain $S$ as a subset as they would not contain the monomial $x$.
So, my guess is that a general ideal generated by $S$ in $R$ would look like$$\mathcal{I}\left(S\right)=RS+SR+RSR$$which is all the ways we can combine $R$ and $S$ as finite linear combinations under the ring operations.
Clearly, this works as a left, right and two-sided ideal since$$R\mathcal{I}\left(S\right)=RS+RSR\subseteq \mathcal{I}\left(S\right),$$$$\mathcal{I}\left(S\right)R=SR+RSR\subseteq \mathcal{I}\left(S\right)$$and$$R\mathcal{I}\left(S\right)R=RSR\subseteq \mathcal{I}\left(S\right).$$
Having said all that, I have seen some sources include the term $\mathbb{Z}S$ in the summand but surely this cannot be the case as $1\in\mathbb{Z}$ and so this would mean $1s=s\in \mathbb{Z}S$ for each $s\in S$ which would imply $S\subseteq \mathbb{Z}S\subseteq \mathbb{Z}S+ RS+SR+RSR$.
What am I missing here?
Best Answer
I'm not sure if this answers your question. If it doesn't, consider it an extended comment.
Definitions are -well- a matter of definition and thus cannot be wrong or true, but they can certainly be weird or counterintuitive.
And I want to suggest that the definitions you provided for the left/right/two-sided ideal generated by a subset of a (non-unital non-commutative) ring are weird and counterintuitive.
If there is any justice in the world, then in any situation where we have some kind of structure $X$ (which can have sub-somethings) with some subset $S \subseteq X$, then the something generated by $X$ should certainly include $S$, in fact it should be the smallest something contained in $X$ that contains $S$.
The definitions you have provided just don't achieve this. Let's consider the commutative non-unital example of $R=\Bbb Z/4\Bbb Z$ with zero multiplication. What should the ideal generated by $\{2\}$ be? According to the definitions you gave, it should be $\{0\}$. I'd argue that it should be $\{0,2\}$. As you mention that you are interested in the question due to quotients, let me ask you: if someone talks about the quotient of $R$ by the ideal generated by $\{2\}$ would you think of $R$ or of $\Bbb Z/2\Bbb Z$ with zero multiplication?
Now I'm not suggesting that we should change the meaning of the notation $RS$ or $SR$ or $RSR$, but I'd suggest that we don't think of those as the (left/right/bilateral) ideal generated by $S$.
In the terminology I'm suggesting, the left ideal generated by $S$ would be $\Bbb ZS+RS$, the right ideal generated by $S$ would be $\Bbb ZS+SR$ and the two-sided ideal generated by $S$ would be $\Bbb ZS+RS+SR+RSR$.
Here's another way to think about this. For any (non-unital) ring $R$, we can define the unitalization $R^+$ which is just $R\times\Bbb Z$ with the product $(r,n)(s,m)=(rs+ns+mr,nm)$. This has the property that (non-unital) left/right/two-sided modules over $R$ correspond to unital left/right/two-sided modules over $R^+$. In particular, $R$ itself is a two-sided module over $R^+$. Now if we take a subset $S \subset R$, we may consider the left/right/two-sided $R^+$-submodule generated by that subset. If we want our equivalence between non-unital $R$-modules and unital $R^+$-modules to respect "the submodule generated by a subset", then we are forced to define them as I did above.