Subrng

A subrng is a subset of a rng $𝑆 \subseteq 𝑅$ such that $𝑆$ is a ring under the same operations, #m/def/ring i.e. $𝑆$ forms both a subgroup under addition and subsemigroup under multiplication of $𝑅$ .

Subrng test

Theorem. Iff $𝑎 - 𝑏$ and $𝑎 𝑏$ are in $𝑆$ whenever $𝑎, 𝑏 \in 𝑆$ , then $𝑆$ is a subrng of $𝑅$ . #m/thm/ring

Proof

The One step subgroup test tests for the additive subgroup, whereas closure is necessary and sufficient for multiplication.

#state/tidy | #lang/en | #SemBr