Semiring Ideal

A left ideal of a Semiring $R$ is a Submonoid $I\subseteq R$ of the additive monoid of $R$ that "absorbs multiplication on the left". Explicitly:

• $0\in I$
• If $x\in I$ and $y\in I$, then $x+y\in I$.
• For every $r:R$ and $x\in I$, $rx\in I$.

Dually, a right ideal of $R$ is a Submonoid $I\subseteq R$ that "absorbs multiplication on the right".

• For every $x\in I$ and $r:R$, $xr\in I$.

A two-sided ideal is a Submonoid $I\subseteq R$ that is both a left and right ideal. When $R$ is Commutative, these three notions all coincide.

Two-sided Ideals and Kernels

There is a 1-1 correspondence between two-sided ideals and the Kernels of Semiring Homomorphisms.

As Modules

Every left ideal $I$ can be thought of as a Submodule of $R$, where $R$ is viewed as a Left Module over itself. Dueally, right ideals can be viewed as Submodules of $R$ viewed as a right module, and two-sided ideals can be viewed as Subbimodules of $R$ viewed as an $(R,R)$ Bimodule.

Categorification

As the previous section suggests, ideals are equivalent to submodules of $R$, where $R$ is viewed as a module over itself in some way. This is akin to viewing $R$ as a sort of "representable module" over itself; from this perspective, an ideal is a submodule of a representable module; EG, a Sieve.