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This is how the wikipedia article on subring defines the subring test

The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and multiplication, and contains the multiplicative identity of $R$.

When you follow the link for the subring test, it is stated as follows

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.

My question is, is the first statement of subring test correct? This is also how a subring is "defined" in Atiyah-Macdonald. It seems incorrect to me as $\mathbb{R}_+$ satisfies those conditions and is not a subring unless I am missing something.

Looking at the responses I feel I should further clarify my question. Closure under subtraction and multiplication (with the added provision that the given subset contain the identity depending on how you define your rings), guarantees a subring, as in the second statement. I am comfortable with this statement as I know that closure under subtraction for a subset of a group (written additively) gives a subgroup. My question is whether the first statement is correct - is closure under addition and multiplication enough?

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    @yoyo: In either case, my question is, is closure under addition and multiplication, enough to guarantee a subring as in the first statement.2011-05-22
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    natural numbers are not a subring of the integers despite being closed under addition and multiplication2011-05-22

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