Let $R$ be a ring (throughout this post, by "ring" I'll mean not necessarily an "associative ring"). If we denote by $+$ the addition component-wise, we get that $(R^n,+)$ is an abelian group ($n$ integer, $n \ge 2$) with additive unit $\bar 0:=(0,0,\dots,0)$. If we further request:
- $R$ with unit $1$;
- $(R^n,+)$ left $R$-module
then $\forall \bar u \in R^n, \bar u = u^i\bar e_i$ (summation convention), where $\bar e_i:=(0,\dots,0,1,0,\dots,0)$ (the only $1$ is in the i-th entry).
Now, suppose that we want to build up in $R^n$ a multiplication, "$\times$", such that:
- $(R^n,+,\times)$ is a ring;
- "$\times$" is bilinear on $R$.
Then, $\bar e_i \times \bar e_j=\alpha_{ij}^k\bar e_k$ completely defines the multiplication under construction.
A wishlist for $(R^n,+,\times)$, and the according constraints on the general $\alpha$s, could start with:
- $(R^n,+,\times)$ associative ring:
$$\alpha_{jk}^l\alpha_{il}^m=\alpha_{ij}^l\alpha_{lk}^m, \forall i,j,k,m \tag 1$$
- $(R^n,+,\times)$ ring with unit $\bar e_1$:
$$\alpha_{1j}^k=\alpha_{j1}^k=\delta_j^k, \forall j,k \tag 2$$
($\delta_l^m$ is Kronecker symbol).
Example: $n=2$. $(R^2,+,\times)$ is a ring with unit $\bar e_1$ if and only if (from $(2)$):
$\alpha:=\alpha_{22}^1, \beta:=\alpha_{22}^2$
$\alpha_{11}^1=\alpha_{12}^2=\alpha_{21}^2=1$
$\alpha_{12}^1=\alpha_{21}^1=\alpha_{11}^2=0$
(For $R=\mathbb{R}$, so loosely constrained, these coefficients should include either complex and split-complex numbers cases, according to the fulfilment or not of additional constraints.)
These coefficients seemingly fulfill $(1)$, so that -at least for $n=2$- $(R^n,+,\times)$ ring with unit $\Rightarrow (R^n,+,\times)$ associative ring
My questions are:
Is the requirement 2. (..."bilinear"...) essential to build up a ring out of $R^n$? (I have the feeling to have requested it just to pop into linear algebra and make things somehow accessible to my background)
Is it true in general that ring with unit implies associative ring?