MatrixNearrings

Matrix nearrings in GAP?

Let $N$ be a (right) nearring and $D=N^n$ a direct product of $N$. For $x\in D$, $x_i$ deonotes the $i$-th component of $D$. Denote by $\iota_j:N\to D$ the $j$-th embedding of $N$ to $D$. An elementary function from $D$ to $D$ is the given by $f_{ij}^r(x)=\iota_j(rx_i)$ for $x\in D$, where $1\leq i\leq n$, $1\leq j\leq n$ and $r\in N$. The matrix nearring $\mathbb M_n(N)$ is the subnearring of the transformation nearring $M(D)$ generated by the set $\{f_{ij}^r\mid 1\leq i\leq n$, $1\leq j\leq n$, $r\in N\}$.

Note that in the above definition, the elementary functions are defined by $f_{ij}^r(x)=\iota_j(rx_i)$ for $x\in D$. If we define $g_{ij}^r(x)=\iota_j(x_ir)$ for $x\in D$, then $g_{ij}^r$ is an endomorphism of $D$.

<b>Theorem.</b> Let $N$ be a simple nearring with $1$. Then the matrix nearring $\mathbb M_n(N)$ is simple.

<b>Theorem</b> (Ke and Wendt). Let $N$ be an integral planar nearring and $n\geq 1$. Then the matrix nearring $\mathbb M_n(N)$ is simple.

In general, $\mathbb M_n(N)\not=M(N)$ even if $N$ is simple with $1$.

<b>Theorem.</b> Let $N$ be a (right) nearring and $\mathbb M_n(N)$ the $n\times n$ matrix nearring on $N$. Then $\mathbb M_n(N)$ has identity if and only if $N$ has a left identity.

<b>Question.</b> Is there a natural way of identifying $\mathbb M_n(N)$ with a sandwitch centralizer nearring when $N$ is a zero symmetric nearring?

<b>Question.</b> Is there a density theorem for $2$-primitive (zero symmetric) nearerings in matrix nearrings $\mathbb M_n(N)$ with $N$ integral planar?

<b>Question.</b> Let $f$ be a bijection in a matrix nearring $\mathbb M_n(N)$. Then $f^{-1}$ exists in $M(N^n)$. When $N$ is finite, a bijection is of finite order, and so $f^{-1}=f^n$ for some positive integer $n$. This makes $f^{-1}$ a member of $\mathbb M_n(N)$. <i>Is it true that $f^{-1}\in \mathbb M_n(N)$ when $N$ is infinite?</i>

Here are the list of Left Nearrings of Small Order Having Right Identities?.