Let $R$ be a non-unital ring, that is, a ring without a multiplicative identity (or a rng if you prefer). If I want to talk about finitely generated modules over $R$, one factible definition is that a (left) $R$-module $M$ is finitely generated if there exist a finite family of elements $m_1,\dots,m_n\in M$ with the property that for all $m\in M$ there exist $r_1,\dots,r_n\in R$ such that $$ m = r_1m_1 + \cdots + r_nm_n. $$
Now, this definition is very bad behaved, in the sense that very simple objects, expected to be in the category of finitely generated modules, fail to be finitely generated:
- As a first example, take $R=2\mathbb{Z}$, then $\mathbb{Z}$ is a $R$-module and its not finitely generated as it is impossible to write $1$ as a $R$-linear combination of elements in $\mathbb{Z}$.
- A second example: Let $k$ be a field and $R=\{f(X)\in k[X]\; | \; f(0)=0\}$, which is a non-unital subring of $k[X]$. Then $k$ has the structure of a $R$-module given by $f(X)\cdot a = 0$ for all $f(X)\in R$ and $a\in k$ and, besides $k$ being one dimensional as a $k$-vector space, it is not finitely generated as a $R$-module.
So, is there another definition of finitely generated module over a non-unital ring that behaves better than this definition?
Another question, what are some examples of finitely generated modules over non-unital rings under the definition given above?
My motivation for asking this question is the following. I'm reading the article On primitive ideals by Victor Ginzburg. In Section 4 he considers the property of a $A$-module to be finitely generated over $A^+$ (where $(A^{\pm},\delta)$ is a triangular structure on $A$), and here $A^+$ is a non-unital subalgebra of $A$. Similarly in Section 5 he defines the category $\mathrm{Mod}(A:\mathcal{U}_+)$ of $A$-modules finitely generated over $\mathcal{U}_+$ where $\mathcal{U}_+$ is a certain non-unital subalgebra o $A$.
A module $M$ over a possibly non-unital ring $R$ is finitely generaed if there exists a finite subsetset $M^{\prime}$ of $M$ such that every element of $M$ is the sum of finitely many elements (possibly 0) of $M^{\prime}$ (repeitions allowed) and a finite $R$-linear combination of $M^{\prime}$.