unity of subring


Let S be a proper subring of the ring R.  If S has a non-zero unity u which is not unity of R, then u is a zero divisorMathworldPlanetmath of R.

Proof.  Because u is not unity of R, there exists an element r of R such that  rur.  Then we have  (ru)u=r(uu)=ru, which implies that  0=(ru)u-ru=(ru-r)u.  Since neither  ru-r  nor  u  is 0, the element  u  is a zero divisor in R.

Title unity of subring
Canonical name UnityOfSubring
Date of creation 2013-03-22 14:49:40
Last modified on 2013-03-22 14:49:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 20-00
Classification msc 16-00
Classification msc 13-00
Related topic UnitiesOfRingAndSubring
Related topic CornerOfARing