The subring test (subtraction vs. addition closure)

Question

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?

Answer 1

Yes, you're right: the version of the subring test found in the wikipedia article on "subring" was faulty, whereas the article subring test has a correct statement.

I just edited the first wikipedia article to read as follows:

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

I hope all will agree that this is an appropriate statement.

There is a slight discrepancy in that the article on subrings explicitly assumes we are working in the category of rings (in which we have a multiplicative identity which all the homomorphisms much respect), whereas the article on the subring test works in the category of rngs (i.e., there may not be a multiplicative identity and even if there is it need not be preserved by homomorphisms). In the category of rngs, one should state the nonemptiness explicitly, whereas in the category of rings it is guaranteed by the presence of the multiplicative identity.

If anyone has further ideas for improving either of these two articles, please let me know. Or rather, please go ahead and implement them -- be bold, as they say on that other site -- but it would be nice to come back here and tell us what you've done.

Answer 2

As your counterexample shows, the first statement is incorrect. But there is a one character fix: require that the subset contains $\,{-}1\,$ vs. $\,1.\,$ Alternatively, require it to also be closed under negation (additive inverses). Perhaps they meant $\rm\,S\,$ is closed under subtraction (vs. addition), since then the subgroup test implies $\rm\,S\,$ is a subgroup of the additive group of $\rm\,R.\,$ Or, explicitly

$$\rm r,s\in S\ \Rightarrow\ r-r\ =\ 0\in S\ \Rightarrow\ 0-r\ =\: -r\in S\ \Rightarrow\ s-(-r)\: =\ s+r\in S $$

Remark $ $ A similar error appears in Greuel and Pfister: A singular introduction to commutative algebra, p. $1$, where they define a subring as a subset containing $1$ that is closed under the induced ring operations. But negation is not explicitly listed as a ring operation in their definition of a ring. Rather, similar to Atiyah and MacDonald, they say that "R, together with addition, is an abelian group". Perhaps "addition" is meant to denote the full additive group structure, so that the ring is supposed to inherit the negation operation (or equivalent axioms for inverses) from the abelian group structure, when it inherits the addition operation.

Answer 3

It seems that the problem lies in what it means to be closed under addition. My interpretation of being closed under addition is that if you restrict the binary operation of addition to the subset that you want to study, then you get a well defined function.

Say, if you have a ring $(R, +, \cdot)$, then if we have a subset $S \subseteq R$, I would interpret $S$ being closed under the addition inherited from $R$ as meaning that if $a, b \in S$ then $a + b \in S$, or that the image of the map

$$+ : S \times S \longrightarrow R$$

that results from restricting the addition to the elements of $S$ lies in $S$, that is, that $+(S \times S) \subseteq S$ (however weird that notation may seem). So if this is what is what it means for $S$ to be closed under addition, then certainly $\mathbb{R}_{+}$ would satisfy the requirements in the first formulation of the "subring test" that you give, but it will not be a subring of $\mathbb{R}$ since it will not contain the additive inverses.

The same thing would happen when considering $\mathbb{N} \subseteq \mathbb{Z}$.

I just checked my copy of Atiyah-Macdonald and indeed they define a subring in this way, so maybe it is just a misunderstanding or maybe a lack of care when defining a subring.