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.
$e(a)e$ is the additive inverse of $eae$ in $eRe$,

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 $e\ne 1$ 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 $f=1e$, then $fRf$ is also a proper corner of $R$.
Remark. If $R$ has 1 with $e\ne 1$ an idempotent. Then corners $S=eRe$ and $T=fRf$, where $f=1e$, 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.
Title  corner of a ring 

Canonical name  CornerOfARing 
Date of creation  20130322 15:43:56 
Last modified on  20130322 15:43:56 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16S99 
Related topic  UnityOfSubring 