Suppose we have ring $R[M]$ over monoid $M$ in real number $R$.The number of generators for the monoid, is finite. Now suppose that every ring element $r$ has decomposition in finite linear basis of n-elements. So ring is subspace of n-dimensional vector space.

**How many rings, depending on n has this property?** Could You give me some references ( books, preprints, articles) when such theorem is stated or proved?

**Motivation:** Here monoid ring and some structure within it - how is it called? in a comment to the question, Scott Carnahan wrote:

The existence of the decomposition of elements of R[M] implies the ring is a subspace of a 4-dimensional real vector space. There are many such rings, but only finitely many monoid rings of dimension less than 5.

**Clarification of the problem:**

I have finitely presented noncommutative monoid with unity and two generators $g_1,g_2$: M = F/Rel where $Rel = \{g_1^2 = e , g_2^2 = g_2 \}$, $e$ is unit element and $F$ is free monoid over two generators. Because of relations $Rel$ every elemet in monoid has form for example $g = g_1g_2g_1g_2...g_1g_2$ ( alternating finite sequence with subscripts 1212... or 2121...). Different monoid elements contains different number of multiplications. It is very simple although infinite multiplicative structure.

Then I consider monoid ring over reals $R[M]$. Every element in $R[M]$ has form:

(1) $t = r_1g_1 + r_2g_2 r_3g_1g_2+r_4g_2g_1+ r_5g_1g_2g_1 + ...+r_p g_ig_kg_i...g_s+ ...$ and so on. $r_i \in R$ and $g_i \in M$.

Note that in general monomial element $g_ig_k...g_s$ every subscript has value in $\{1,2\}$ and no two following each other subscripts are the same ( they alternate like in sequence like $1212..$ or $2121..$. Of course this is standard ring definition.

In structure, I would like to describe You here, I have strange additional property: there is element $g_3$ in ring $R[M]$ ( **but it is not monoid element!**) which allows following decomposition:

For every $r \in R[M]$ we have

$r= r_0 e + r_1 g_1 +r_2 g_2 +r_3 g_3$

Look: there are **only four terms in decomposition, even if You decompose general ring element in the form of (1)**. However after such decomposition I may **only multiply** such elements and **not add them**. In some way it looks like vector space. From the other side such decomposition may be treated as another monoid. So in fact decomposition as above, I trying to treat as some kind of "parametrization" of ring elements. Indeed it is element of some vector space, but there are further requirements on coefficients $r_i$ ( which I do not need to describe here, they in fact are part of monoid definition).

As far as I know this is not standard ring property - maybe I am wrong. If I think about for example polynomial ring (that in simple case is real ring over multiplicative monoid generated by one generator $x$) such decomposition is not possible.