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# Subring test

In abstract algebra, the subring test is a theorem that states that for any ring, a nonempty subset of that ring is a subring if it is closed under multiplication and subtraction. Note that here that the terms ring and subring are used without requiring a multiplicative identity element.

More formally, let $R$ be a ring, and let $S$ be a nonempty a subset of $R$. If for all $a, b in S$ one has $ab in S,$ and for all $a, bin S$ one has $a - b in S,$ then $S$ is a subring of $R$.

If rings are required to have unity, then it must also be assumed that the multiplicative identity is in the subset.

## Proof

Since $S$ is nonempty and closed under subtraction, by the subgroup test it follows that $S$ is a group under addition. Hence, $S$ is closed under addition, addition is associative, $S$ has an additive identity, and every element in $S$ has an additive inverse.

Since the operations of $S$ are the same as those of $R,$ it immediately follows that addition is commutative, multiplication is associative, multiplication is left distributive over addition, and multiplication is right distributive over addition.

Thus, $S$ is a subring of $R$.

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