[[Ring theory MOC]]
# Subrng

A **subrng** is a subset of a [[rng]] $S \sube R$ 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 $ab$ are in $S$ whenever $a,b \in S$, then $S$ is a subrng of $R$. #m/thm/ring

> [!check]- Proof
> The [[Subgroup#One step subgroup test]] tests for the additive subgroup, whereas closure is necessary and sufficient for multiplication.
> <span class="QED"/>

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