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# corner of a ring

Does there exist a subset $S$ of a ring $R$ which is a ring with a multiplicative identity, but not a subring of $R$?

Let $R$ be a ring without the assumption that $R$ has a multiplicative identity. Further, assume that $e$ is an idempotent of $R$. Then the subset of the form $eRe$ is called a *corner* of the ring $R$.

It’s not hard to see that $eRe$ is a ring with $e$ as its multiplicative identity:

1. $eae+ebe=e(a+b)e\in eRe$,

2. $0=e0e\in eRe$,

3. 4. $(eae)(ebe)=e(aeb)e\in eRe$, and

5. $e=ee=eee\in eRe$, with $e(eae)=eae=(eae)e$, for any $eae\in eRe$.

If $R$ has no multiplicative identity, then any corner of $R$ is a proper subset of $R$ which is a ring and not a subring of $R$. If $R$ has 1 as its multiplicative identity and if $e\neq 1$ is an idempotent, then the $eRe$ is not a subring of $R$ as they don’t share the same multiplicative identity. In this case, the corner $eRe$ is said to be *proper*. If we set $f=1-e$, then $fRf$ is also a proper corner of $R$.

Remark. If $R$ has 1 with $e\neq 1$ an idempotent. Then corners $S=eRe$ and $T=fRf$, where $f=1-e$, are direct summands (as modules over $\mathbb{Z}$) of $R$ via a Peirce decomposition.

# References

- 1
I. Kaplansky,
*Rings of Operators*, W. A. Benjamin, Inc., New York, 1968.

## Mathematics Subject Classification

16S99*no label found*

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## Comments

## Rings with corners?

Hmmm .... rings with corners? Sounds like just the thing for people with square fingers :)

## Re: Rings with corners?

It's neat to see mathematicians use humor sometimes in creating names for mathematical constructs. I found it in Rings of Operators by the great algebraist of the 20th century Irving Kaplansky, and I will add the reference to the entry. The term is also used by my former prof T. Y. Lam.

If anyone has come across a funny mathematical term, please share with the rest of us by creating an entry for it.

## Re: Rings with corners?

>If anyone has come across a funny mathematical term, please share with >the rest of us by creating an entry for it.

One favorite is "earthquake measure". See

http://www.math.sunysb.edu/~saric/research/ea12.pdf

Google has a lot of references on this one too.

## Re: Rings with corners?

> If anyone has come across a funny mathematical term, please

> share with the rest of us by creating an entry for it.

Though it's not really a funny term, the term "adele" is another example of whimsical mathematical nomenclature...the term "idele" comes from the notion of generalizing an "ideal", and adeles were an additive version of ideles, hence the name.

Cam