surjective homomorphism between unitary rings

Theorem.  Let f be a surjectivePlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from a unitary ring R to another unitary ring R.  Then

Proof.1.  In a ring, the identity element is unique, whence it suffices to show that f(1) has the properties required for the unity of the ring R.  When a is an arbitrary element of this ring, there is by the surjectivity an element a of R such that  f(a)=a.  Thus we have


2.  Let a be a unit of R.  Then

f(a)f(a-1)=f(aa-1)=f(1)= 1,f(a-1)f(a)=f(a-1a)=f(1)= 1,

whence f(a-1) is a multiplicative inverse of f(a).

Title surjective homomorphism between unitary rings
Canonical name SurjectiveHomomorphismBetweenUnitaryRings
Date of creation 2013-03-22 19:10:22
Last modified on 2013-03-22 19:10:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 16B99
Classification msc 13B10
Related topic IsomorphismSwappingZeroAndUnity