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.
Best Answer
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.