Semiring Extension

A semiring extension of a semiring R 𝑅 R by a Commutative Monoid I 𝐼 I is a Semiring E 𝐸 E and a Semiring Homomorphism ϕ : E → R : italic-ϕ → 𝐸 𝑅 \phi:E\to R that fit into the following Short Exact Sequence of Commutative Monoids:

0→I→E→ϕR→0→0𝐼→𝐸italic-ϕ→𝑅→0\par 0\to I\to E\xrightarrow{\phi}R\to 0

This turns I 𝐼 I into a 2-Sided Ideal of E 𝐸 E .

Via Displayed Semirings

If we view the pair ( E , ϕ ) 𝐸 italic-ϕ (E,\phi) as a Displayed Semiring over R 𝑅 R , then the map I → E → 𝐼 𝐸 I\to E can be viewed as a Monoid Homomorphism from I 𝐼 I into the Kernel E 0 subscript 𝐸 0 E_{0} of E ⇾ R ⇾ 𝐸 𝑅 E\rightarrowtriangle R .