Subring test

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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 one has and for all one has 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.

[edit] 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.