Name for a "subring without identity with identity"

Question

Suppose you have the ring $\Bbb R^2$. The set of all elements of the form $(x,0)$ forms its own ring, which is a unital ring with multiplicative identity element $(1,0)$. However, this is not a "subring" of $\Bbb R^2$ because it doesn't have the same $1$, so the embedding is not a "ring homomorphism," at least within the textbooks I've been reading.

We could view this as a "subrng" of $\Bbb R^2$. But the terminology is kind of weird, because it's a "subrng which is also a ring," but not a "subring." Put another way, it's a "subring without identity with identity."

Is there any standard terminological name for this? A "unital subrng?" A "non-unital subring with unity?" etc? A "sub-ring-not-necessarily-with-unity-but-which-has-unity-anyway?"

This question is purely about terminology and is different from other related questions asked, such as this and this and this. I would just like to know what term to use for these.

EDIT: it seems that a few people have suggested "corner ring," but corner rings seem to be different: they basically involve multiplying the entire ring by some idempotent. Any corner ring will also be a "unital subring", but any proper subring of a corner ring won't be a corner ring of the original ring. For instance, $(z,0)$ with $z \in \Bbb Z$ isn't a corner ring of $\Bbb R²$, but it is a "unital subrng" of it.

Answer 1

These are the subrings of corner rings. From Wikipedia:

"If $a$ is idempotent in the ring $R$, then $aRa$ is again a ring, with multiplicative identity $a$. The ring $aRa$ is often referred to as $a$ corner ring of $R$. The corner ring arises naturally since the ring of endomorphisms $\operatorname{End}_R(aR) \cong aRa$"

This is quite special, though. You can think of other situations as well where a subobject with respect to a different category can be casted into an object of the given category. For example, a Banach space with a subspace of the underlying vector space which becomes (only) a Banach space when equipped with a different norm (think of $\ell^p \subseteq \ell^q$ for $p<q$). Or a boolean algebra with a subset of the underlying set that becomes a boolean algebra with different operations (the boolean algebra of regular open subsets of a topological space is a prominent example). I don't think that there will be already a general name for this phenomenon.

Answer 2

The ring $(R,+,\cdot,(0,0),\hat{1})$, with $R = \{(x,y) \in \mathbb{R}^2 \,| \, x \in \mathbb{R}\, y = 0 \}$, $+,\cdot$ the usual operators in $\mathbb{R}^2$ and $\hat{1} := (1,0) \in \mathbb{R}^2$, is a subset of $\mathbb{R}^2$ but it's not a subring of $(\mathbb{R}^2,+,\cdot,(0,0),(1,1))$.

So $R$ is just a subset of $\mathbb{R}^2$ that happens to be a ring, in other words $R$ is just a ring contained in $\mathbb{R}^2$. I don't know if there is a terminology for this scenario, but we could just say that "$R$ is a ring subset of $\mathbb{R}^2$".

Edit

The comment of Geoffrey Trang made me do some research and I found the following property:

If $a$ is idempotent in the ring $S$, then $aSa$ is again a ring, with multiplicative identity $a$. The ring $aSa$ is often referred to as a corner ring of $S$.

So it turns out that in this case your ring $R = \{(x,y) \in \mathbb{R}^2 \,| \, x \in \mathbb{R}\, y = 0 \}$ can be called corner ring of $\mathbb{R}^2$. Becasue, $(1,0)$ is idempotent in the ring $(\mathbb{R}^2,+,\cdot,(0,0),(1,1))$, and

$$(1,0)\mathbb{R}^2(1,0) = \{(1,0)x(1,0) \, | \, x \in \mathbb{R}^2\} = R.$$

Answer 3

As we see here, you'll have people say "it's not a subring." And if a ring doesn't have identity they'll say "it's not a ring." But frankly these proclamations are silly.

It's silly because terminology isn't some fixed and unbending universal thing passed down from heaven writ in stone. You don't have to fret about choosing "the perfect" or "right" term for something for fear of "having wrong terminology."

If you are writing something up, one which will use the notion over and over again, you can use whatever you think is clear and that the reader can come to terms with easily. If you need to discuss "nonunital subrings with identity" in something you are writing up, you simply say "the term subring will be used to mean (...stuff...) with identity, but not necessarily sharing the same identity as the containing ring." You just have to do your best to use something familiar and hopefully not awkward.

Authors (good ones) have used the word subring to talk about non-unital subrings. And they have also used the word "ring" to mean a ring not necessarily with identity, and the world survived.

You have license to flex terminology at least that much. Use your common sense and don't use wild terminology that makes your exposition hard to read.