corner of a ring

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

Let R be a ring without the assumptionPlanetmathPlanetmath that R has a multiplicative identity. Further, assume that e is an idempotentPlanetmathPlanetmath 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. 1.


  2. 2.


  3. 3.

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

  4. 4.

    (eae)(ebe)=e(aeb)eeRe, and

  5. 5.

    e=ee=eeeeRe, with e(eae)=eae=(eae)e, for any eaeeRe.

If R has no multiplicative identity, then any corner of R is a proper subsetMathworldPlanetmathPlanetmath of R which is a ring and not a subring of R. If R has 1 as its multiplicative identity and if e1 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 e1 an idempotent. Then corners S=eRe and T=fRf, where f=1-e, are direct summandsMathworldPlanetmath (as modules over ) of R via a Peirce decompositionMathworldPlanetmath.


  • 1 I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York, 1968.
Title corner of a ring
Canonical name CornerOfARing
Date of creation 2013-03-22 15:43:56
Last modified on 2013-03-22 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