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Ring homomorphism

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A mapping from a ring Rto a ring S that preserves thetwo ring operations; that is, for all ,a b R , we have

( ) ( ) ( )a b a b and ( ) ( ) ( )ab a b .

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Ring isomorphism

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A ring homomorphism that is both one-to-one and onto.

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Properties of ring homomorphisms

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Let be a ring homomorphism from a ring Rto a

ring S. LetA be a subring ofRand B an ideal ofS.

1. For any r R and any positive integern,

( ) ( )nr n r and ( ) ( ( ))n nr r .

2. ( ) { ( ) | }A a a A is a subring ofS.

3. IfA is an ideal and is onto S, then ( )A is anideal.

4. 1( ) { | ( ) }B r R r B is an ideal ofR.

5. IfRis commutative, then ( )R is commutative.

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More properties of ring homomorphisms

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Let be a ring homomorphism from a ring Rto a ring

S. LetA be a subring ofRand B an ideal ofS.

IfRhas a unity 1, {0}S , and is onto, then(1) is the unity ofS.

7. is an isomorphism if and only if is onto and

{ | ( ) 0} {0}Ker r R r .

8. If is an isomorphism from Ronto S, then 1 is

an isomorphism from S onto R.

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Kernels are Ideals

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Let be a homomorphism from a ring Rto a ring S.

Then { | ( ) 0}Ker r R r is an ideal ofR.

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First Isomorphism Theorem for Ringsa.k.a.

Fundamental Theorem of Ring Homomorphisms

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Let be a ring homomorphism from Rto S. Then the

mapping from /R Ker to ( )R , given by

( )r Ker r , is an isomorphism. In symbols,

/ ( )R Ker R .

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Ideals are Kernels

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Every ideal of a ring Ris the kernel of a ringhomomorphism ofR. In particular, an idealA is the

kernel of the mapping r r A from R to /R A .

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Natural homomorphism from Rto R/A

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The mapping r r A .

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Homomorphism from Zto a Ring with Unity

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Let Rbe a ring with unity 1. The mapping :Z R

given by 1n n is a ring homomorphism.

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A Ring with Unity Contains nZ orZ

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IfRis a ring with unity and the characteristic ofRis0n , then Rcontains a subring isomorphic to nZ . If

the characteristic ofRis 0, then Rcontains a subringisomorphic to Z.

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mZ is a Homomorphic Image ofZ

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For any positive integerm, the mapping of : mZ Z

given by modx x m is a ring homomorphism.

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A Field Contains pZ orQ

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IfFis a field of characteristicp, then Fcontains asubfield isomorphic to pZ . IfFis a field of

characteristic 0, then Fcontains a subfield isomorphicto the rational numbers.

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Prime subfield

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The smallest subfield (a subfield contained in everysubfield).

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Field of Quotients

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Let D be an integral domain. Then there exists a field F(called the field of quotients ofD) that contains a

subring isomorphic to D.