unity of subring

Theorem.

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 divisor of $R$.

Proof.  Because $u$ is not unity of $R$, there exists an element $r$ of $R$ such that  $ru\ne r$.  Then we have  $(ru)u=r(uu)=ru$, which implies that  $0=(ru)u-ru=(ru-r)\cdot u$.  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