📂Abstract Algebra

Definition and Criterion of Subrings

Definition ¹

A subset $S$ of a ring $R$ is called a subring$\mathrm{Subring}$ of $R$ if it satisfies the conditions of a ring with respect to the operations of $R$.

Meanwhile, it is trivial that $\left\{ 0 \right\}$ and $R$ are subrings of the ring $R$, hence $\left\{ 0 \right\}$ and $R$ are referred to as trivial subrings ($\mathrm{trivial\ subring}$).

Theorem: Subring Criterion

For a non-empty subset $S$ of a ring $R$, if whenever $a,\ b$ is an element of $S$, $a-b,\ ab$ is also an element of $S$, then $S$ is a subring of $R$. That is, if $S$ is closed under subtraction and multiplication, it is a subring of $R$.

Proof

Assume that when $a,\ b$ is an element of the subset $S$, $a-b,\ ab$ is also an element of $S$.

1. By assumption, according to the subgroup criterion, $S$ is a group under addition.
2. The operations of $S$ are the same as those of the ring $R$, so it is trivially commutative.
3. It is also trivially closed under multiplication by assumption.
4. Since the operations of subset $S$ are the same as those of the ring $R$, it is also trivial that the associative law for multiplication holds.
5. For the same reasons, it is natural that the distributive laws for addition and multiplication hold within the subset $S$.

From 1 to 5, since the subset $S$ is closed under both operations, is an abelian group under addition, and satisfies the associative law for multiplication, and the distributive laws for both addition and multiplication, $S$ is a ring. Therefore, the subset $S$ is a subring of $R$.

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