# Subring Test

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## Theorem

Let $S$ be a subset of a ring $\struct {R, +, \circ}$.

Then $\struct {S, +, \circ}$ is a subring of $\struct {R, +, \circ}$ if and only if these all hold:

- $(1): \quad S \ne \O$
- $(2): \quad \forall x, y \in S: x + \paren {-y} \in S$
- $(3): \quad \forall x, y \in S: x \circ y \in S$

## Proof

### Necessary Condition

If $S$ is a subring of $\struct {R, +, \circ}$, the conditions hold by virtue of the ring axioms as applied to $S$.

### Sufficient Condition

Conversely, suppose the conditions hold. We check that the ring axioms hold for $S$.

- $(1): \quad$
**Ring Axiom $\text A$: Addition forms an Abelian Group:**By $1$ and $2$ above, it follows from:- that $\struct {S, +}$ is an abelian subgroup of $\struct {R, +}$, and therefore an abelian group.

- $(2): \quad$
**Ring Axiom $\text M0$: Closure under Product:**From $3$, $\struct {S, \circ}$ is closed. - $(3): \quad$
**Ring Axiom $\text M1$: Associativity of Product:**From Restriction of Associative Operation is Associative, $\circ$ is associative on $R$, therefore also associative on $S$. - $(4): \quad$
**Ring Axiom $\text D$: Distributivity of Product over Addition:**From Restriction of Operation Distributivity, $\circ$ distributes over $+$ for the whole of $R$, therefore for $S$ also.

So $\struct {S, +, \circ}$ is a ring, and therefore a subring of $\struct {R, +, \circ}$.

$\blacksquare$

## Also defined as

Some sources insist that for $\struct {S, +, \circ}$ to be a subring of $\struct {R, +, \circ}$, the unity (if there is one) must be the same for both, but this is an extra condition as this is not necessarily the general case.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Theorem $21.4$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 19$. Subrings: Theorems $31, \ 32$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.1$: Subrings: $2.2$ Lemma - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 56.2$ Subrings and subfields