Square-Zero Extension

The square-zero extension of a Semiring $R$ by an $(R,R)$ -Bimodule $M$ is a semiring structure on the Tensor Product of Bimodules $R\oplus M$ where

Addition is given by addition in $R\oplus M$
Multiplication $(r_{1},m_{1})\cdot(r_{2},m_{2})$ is given by

$(r_{1}r_{2},r_{1}\cdot m_{2}+m_{1}\cdot r_{2})$

First, observe that the pair $(0,0)$ is an Absorbing Element

	$\displaystyle(0,0)\cdot(r,m)$	$\displaystyle=(0r,0\cdot m+0\cdot r)$
		$\displaystyle=(0,0)$

	$\displaystyle(r,m)\cdot(0,0)$	$\displaystyle=(r0,r\cdot 0+m\cdot 0)$
		$\displaystyle=(0,0)$

Moreover, a tedious computation shows that multiplication is Associative

$\displaystyle(r,x)\cdot((s,y)\cdot(t,z))$	$\displaystyle=(r,x)\cdot(st,sz+yt)$	$\displaystyle(\text{by defn.})$
	$\displaystyle=(r(st),r(sz+yt)+x(st))$	$\displaystyle(\text{by defn.})$
	$\displaystyle=(r(st),r(sz)+r(yt)+x(st))$	$\displaystyle(\text{$\cdot$ distributes over $+$})$
	$\displaystyle=((rs)t,(rs)z+r(yt)+(xs)t)$	$\displaystyle(\text{joint associativity})$
	$\displaystyle=((rs)t,(rs)z+(ry)t+(xs)t)$	$\displaystyle(\text{compatibility})$
	$\displaystyle=((rs)t,(rs)z+(ry+xs)t)$	$\displaystyle(\text{$\cdot$ distributes over $+$})$
	$\displaystyle=(rs,ry+xs)\cdot(t,z)$	$\displaystyle(\text{by defn.})$
	$\displaystyle=((r,x)\cdot(s,y))\cdot(t,z)$	$\displaystyle(\text{by defn.})$

Finally, we can validate that multiplication distributes over addition

$\displaystyle(r,x)\cdot((s,y)+(t,z))$	$\displaystyle=(r,x)\cdot(s+t,y+z)$	$\displaystyle(\text{by defn.})$
	$\displaystyle=(r(s+t),r\cdot(y+z)+x\cdot(s+t))$	$\displaystyle(\text{by defn.})$
	$\displaystyle=(rs+rt,ry+rz+xs+xt)$	$\displaystyle(\text{distribute})$
	$\displaystyle=(rs+rt,(ry+xs)+(rz+xt))$	$\displaystyle(\text{commute})$
	$\displaystyle=(r,x)\cdot(s,y)+(r,x)\cdot(t,z)$	$\displaystyle(\text{by defn.})$

$\displaystyle((r,x)+(s,y))\cdot(t,z)$	$\displaystyle=(r+s,x+y)\cdot(t,z)$	$\displaystyle(\text{by defn.})$
	$\displaystyle=((r+s)t,(r+s)z+(x+y)t)$	$\displaystyle(\text{by defn.})$
	$\displaystyle=(rt+st,rz+sz+xt+yt)$	$\displaystyle(\text{distribute})$
	$\displaystyle=(rt+st,(rz+xt)+(sz+yt))$	$\displaystyle(\text{commute})$
	$\displaystyle=(r,x)\cdot(t,z)+(s,y)\cdot(t,z)$	$\displaystyle(\text{by defn.})$

As a Grothendieck Construction

The square-zero extension is a sort of Grothendieck Construction, and gives rise to an equivalence between the Category of Additive Bundles $\mathrm{CMon}(\mathrm{Rig}/R)$ and the Category of (R,R) Semiring Bimodules.

How can we handle iterated square-zero extensions; we will need a way to extend an $(R,R)$ -bimodule $M$ to an $(R\oplus M,R\oplus M)$ bimodule; this amounts to finding the red arrow in the following diagram

which is universally given by the Restriction of Scalars $p^{*}Mp^{*}$ .