The subring test

This is just how the Wikipedia write-up on subring specifies the subring examination

The subring examination states that for any kind of ring $R$ , a nonempty part of $R$ is a subring if it is shut under enhancement and also reproduction, and also has the multiplicative identification of $R$ .

When you adhere to the link for the subring test, it is mentioned as adheres to

In abstract algebra, the subring examination is a theory that mentions that for any kind of ring, a nonempty part of that ring is a subring if it is shut under reproduction and also reduction. Keep in mind that below that the terms ring and also subring are made use of without calling for a multiplicative identification component.

My inquiry is, is the first declaration of subring examination deal with? This is additionally just how a subring is "defined" in Atiyah - Macdonald. It appears wrong to me as $\mathbb{R}_+$ satisfies those problems and also is not a subring unless I am missing out on something.

Considering the feedbacks I feel I need to better clarify my inquiry. Closure under reduction and also reproduction (with the included stipulation that the offered part have the identification relying on just how you specify your rings), assures a subring, as in the 2nd declaration. I fit with this declaration as I recognize that closure under reduction for a part of a team (created additively) offers a subgroup. My inquiry is whether the first declaration is proper - is closure under enhancement and also reproduction sufficient?

2022-06-07 14:31:30
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Answers: 3

It appears that the trouble hinges on what it suggests to be shut under enhancement. My analysis of being shut under enhancement is that if you limit the binary procedure of enhancement to the part that you intend to research, after that you get a well specified function.

Claim, if you have a ring $(R, +, \cdot)$, after that if we have a part $S \subseteq R$, I would certainly analyze $S$ being shut under the enhancement acquired from $R$ as suggesting that if $a, b \in S$ after that $a + b \in S$, or that the photo of the map

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

that arises from limiting the enhancement to the components of $S$ hinges on $S$, that is, that $+(S \times S) \subseteq S$ (nonetheless unusual that symbols might appear). So if this is what is what it suggests for $S$ to be shut under enhancement, after that absolutely $\mathbb{R}_{+}$ would certainly please the needs in the first solution of the "subring test" that you offer, yet it will certainly not be a subring of $\mathbb{R}$ given that it will certainly not have the additive inverses.

The very same point would certainly take place when taking into consideration $\mathbb{N} \subseteq \mathbb{Z}$.

I simply examined my duplicate of Atiyah - Macdonald and also without a doubt they specify a subring this way, so possibly it is simply a misconception or possibly an absence of treatment when specifying a subring.

2022-06-07 14:54:35

Yes, you are appropriate: the variation of the subring examination located in the wikipedia write-up on "subring" was damaged, whereas the write-up subring test has a proper declaration.

I simply modified the first wikipedia write-up to read as adheres to:

"The subring examination states that for any kind of ring R, a part of R is a subring if it has the multiplicative identification of R and also is shut under reduction and also reproduction.'

I wish all will certainly concur that this is an ideal declaration.

There is a mild disparity because the write-up on subrings clearly thinks we are operating in the group of rings (in which we have a multiplicative identification which all the homomorphisms much regard), whereas the write-up on the subring examination operates in the group of rngs (i.e., there might not be a multiplicative identification and also also if there is it need not be maintained by homomorphisms). In the group of rngs, one need to mention the nonemptiness clearly, whereas in the group of rings it is assured by the visibility of the multiplicative identification.

If any person has better suggestions for boosting either of these 2 write-ups, please allow me recognize. Or instead, please proceed and also implement them - - be vibrant , as they claim on that particular various other website - - yet it would certainly behave ahead back below and also inform us what you've done.

2022-06-07 14:52:06

As your counterexample reveals, the first declaration is wrong. Yet there is a one personality solution: call for that the part has $\,{-}1\,$ vs. $\,1.\,$ Alternatively, need it to additionally be shut under negation (additive inverses). Probably they suggested $\rm\,S\,$ is shut under reduction (vs. enhancement), ever since the subgroup test indicates $\rm\,S\,$ is a subgoup of the additive team of $\rm\,R.\,$ Or, clearly

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

Remark $ $ A comparable mistake shows up in Greuel and also Pfister: A singular introduction to commutative algebra, p. $1$ , where they specify a subring as a part having $1$ that is shut under the generated ring procedures. Yet negation is not clearly detailed as a ring procedure in their definition of a ring. Instead, comparable to Atiyah and also MacDonald, they claim that "R, along with enhancement, is an abelian group". Probably "addition" is suggested to represent the complete additive team framework, to make sure that the ring is intended to acquire the negation procedure (or equal axioms for inverses) from the abelian team framework, when it acquires the enhancement procedure.

2022-06-07 14:48:28