Semiring Ideal

Let $R$ be a Ring. A left ideal of $R$ is a Subgroup $I\subseteq R$ of the additive group 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 Subgroup of $I\subseteq R$ that "absorbs multiplication on the right"; EG:

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

A two-sided ideal is a subset $I\subseteq R$ that is both a left and right ideal. When $R$ is Commutative, left, right, and two-sided ideals coincide.

Two-sided Ideals, Kernels, and Congruences

There is a 1-1-1 correspondence between

1. Two-sided ideals $I$
2. The Kernels of Ring Homomorphisms
3. Congruences of rings

• $(1\Leftrightarrow 3)$ Even in a Semiring, we have a bridge between ideals and congruences; we can construct a congruence from an ideal $I$ by forming its Bourne Congruence, and we can construct an ideal from a congruence $\sim$ by forming the ideal $\left\{\;x:R\right\}$ of elements that will lie in the kernel of the inclusion into the Quotient Semiring of $\sim$. However, this correspondence is not an equivalence for semirings; we do not end up reconstructing the same congruence.

  Consider what happens when we round-trip a congruence $\sim$; the resulting congruence $x\sim_{0}y=\exists a,b.\;a\sim 0\land b\sim 0\land x+a=y+b$

As Modules

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

Categorification

If we view $R$ as a single-object Abelian Category, then an ideal $I$ is a Sieve on $A$ that is closed under the additive structure of $R$.

1Lab

• Algebra.Ring.Ideal