Given a ring (with $1$) $R$, one defines what, say, a left ideal is. There's also a natural definition of ideal generated by a subset

Definition A: $_R(S):=\bigcap\{I\supseteq S:I\text{ is a left ideal}\}$

Also, one can define (for the purpose of this question) the following left ideal - "$\langle\,\rangle$" stands for the generated additive subgroup:

Definition B: $RS:=\langle rs\,|\,r\in R\wedge s\in S\rangle$

And it holds

Characterisation A: $_R(S)=RS$

Now, these definitions can be extended to pseudo-rings (a.k.a. rngs, a.k.a. rings without unity). Many basic results can be carried over to this case without sensible effort.

One that cannot is Charactersation A. Indeed, that is false. In this notation it would hold instead $$_R(S)=\langle x,rs\,|\,r\in R\wedge s\in S\rangle=\langle S\cup RS\rangle$$ and, for instance, $S\not\subseteq RS$ in quite the number of cases.

Nontheless, I thought that the standard definition of ideal generated by $S$ was Definition A, for instance, as it is suggested by English Wikipedia.

Recently, I happened to have an argument with a person who claimed that the actual definition in that case was Definition B. We could not discuss the details, though.

I tried doing some research, but what I found did not develop otstandingly this specifical question. Can someone provide me with the correct definition and, in case it is Definition B, some insight on why it is preferred? A reference in a book works as well.

Thank you all in advance

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    $\begingroup$ Calling $RS$ the "ideal generated by $S$" is pretty bad, since it's not guaranteed to contain $S$. $\endgroup$ – Slade May 6 '15 at 18:21
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    $\begingroup$ I've deleted "non-empty" (in the introduction of S) since this is not necessary (in fact, such assumptions appear very often but are more confusing than clarifiying). $\endgroup$ – Martin Brandenburg May 6 '15 at 18:24
  • $\begingroup$ @MartinBrandenburg you did well. $\endgroup$ – user228113 May 6 '15 at 18:28

In my opinion, Definition A is the correct one. ${}_R (S)$ should be the smallest left ideal of $R$ containing $S$. It is additively generated $S$ and "$R \cdot S$". Of course you don't need $-S$ since $r(-s)=-rs$. Even for unital rings, Definition B is not a definition, it should be regarded as a characterization or construction.

Wikipedia phrases it as follows:

If R does not have a unit, then the internal descriptions [of generated ideals] above must be modified slightly. In addition to the finite sums of products of things in X with things in R, we must allow the addition of n-fold sums of the form x+x+...+x, and n-fold sums of the form (−x)+(−x)+...+(−x) for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous.

  • $\begingroup$ Yeah, that was silly of me. I'll edit it, thanks. $\endgroup$ – user228113 May 6 '15 at 18:22

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